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ADVANCED  CALCULUS 


A  TEXT  UPON  SELECT  PARTS  OF  DIFFERENTIAL  CAL- 
CULUS, DIFFERENTIAL  EQUATIONS,  INTEGRAL 
CALCULUS,  THEORY   OF   FUNCTIONS, 
WITH  NUMEROUS  EXERCISES 


BY 


EDWIN  BIDWELL  WILSON,  Ph.D. 

I'ltOl-KSSOU   OF    MATIIKMATICAL  PHYSICS   IN    THE    MASSA*  iUSKTTS 
INSTITITE   OF   TECHNOLOGY 


GINN  AND  COMPANY 

ISOSTOX     •     XK\V    VOKK     •     CHICAGO     •     LOXDOX 
ATr,ANTA     •     DALLAS     •    COLUMBUS     •    SAX   FKAXCISCO 


COrVRIGHT,  1911,  1912,  BY 
EDWIN  BIDWKLL  WILSON 


ALL    RIGHTS    RiiSERVED 
121. G 


GINN  AND  COMPANY  ■  PRO- 
PKIHTORS  •  BOSTON  •  U.S.A. 


e  A  N  T  A  t-  *  ■■■  ^=  •■    "  ■    ■ 


PREFACE 

It  is  probable  that  ahnost  every  teacher  of  advanced  calculus  feels  the 
need  of  a  text  suited  to  present  conditions  and  adaptable  to  his  use.  To 
write  such  a  book  is  extremely  difficult,  for  the  attainments  of  students 
who  enter  a  second  course  in  calculus  are  different,  their  needs  are  not 
uniform,  and  the  viewpoint  of  their  teachers  is  no  less  varied.  Yet  in 
view  of  the  cost  of  time  and  money  involved  in  producing  an  Advanced 
Calculus,  in  proportion  to  the  small  number  of  students  who  will  use  it, 
it  seems  that  few  teachers  can  afford  the  luxury  of  having  their  own 
text ;  and  that  it  consequently  devolves  upon  an  author  to  take  as  un- 
selfish and  unprejudiced  a  view  of  the  subject  as  possible,  and,  so  far  as 
in  him  lies,  to  produce  a  book  which  shall  have  the  maximum  iiexibility 
and  adaptability.  It  was  the  recognition  of  tliis  duty  that  has  kept  the 
present  work  in  a  perpetual  state  of  growth  and  modification  during 
five  or  six  years  of  composition.  Every  attempt  lias  been  made  to  write 
in  such  a  manner  that  the  individual  teacher  may  feel  the  minimum 
embarrassment  in  picking  and  choosing  what  seems  to  him  best  to  meet 
the  needs  of  any  particular  class. 

As  the  aim  of  the  book  is  to  be  a  working  text  or  laboratory  manual 
for  classroom  use  rather  than  an  artistic  treatise  on  analysis,  especial 
attention  has  been  given  to  the  })reparation  of  numerous  exercises  which 
should  range  all  the  way  from  those  which  require  nothing  but  substi- 
tution in  certain  formulas  to  those  which  embody  important  results 
withheld  from  the  text  for  the  purpose  of  leaving  the  student  some 
vital  bits  of  mathematics  to  develop.  It  has  been  fully  recognized  that 
for  the  student  of  mathematics  the  work  on  advanced  calculus  falls  in 
a  period  of  transition,  —  of  adolescence,  —  in  w^hich  he  must  grow  from 
close  reliance  upon  his  book  to  a  large  reliance  upon  himself.  More- 
over, as  a  course  in  advanced  calculus  is  the  ultima  Thule  of  the 
mathematical  voyages  of  most  students  of  physics  and  engineering,  it 
is  a])propriate  that  the  text  placed  in  the  hands  of  those  who  seek  that 
goal  should  by  its  method  cultivate  in  them  the  attitude  of  courageous 

iii 


iv  PKEFACE 

explorers,  and  in  its  extent  supply  not  only  their  immediate  needs,  but 
much  that  may  be  useful  for  later  reference  and  independent  study. 

With  the  large  necessities  of  the  physicist  and  the  growing  require- 
ments of  the  engineer,  it  is  inevitable  that  the  great  majority  of  our 
students  of  calculus  should  need  to  use  their  mathematics  readily  and 
vigorously  rather  than  with  hesitation  and  rigor.  Hence,  although  due 
attention  has  been  paid  to  modern  questions  of  rigor,  the  chief  desire 
has  been  to  conhrm  and  to  extend  the  student's  working  knowledge  of 
those  great  algorisms  of  mathematics  which  are  naturally  associated 
wnth  the  calculus.  That  the  compositor  should  have  set  ^' vigor"  where 
"rigor"  was  written,  might  appear  more  amusing  were  it  not  for  the 
suggested  antithesis  that  there  may  be  many  who  set  rigor  where  vigor 
should  be. 

As  I  have  had  practically  no  assistance  witli  either  tlie  manuscript 
or  the  proofs,  I  cannot  expect  that  so  large  a  Avork  shall  be  free  from 
errors :  I  can  only  have  faith  that  such  errors  as  occur  may  not  prove 
seriously  troublesome.  To  spend  upon  this  book  so  much  time  and 
energy  which  could  have  been  reserved  with  keener  pleasure  for  vari- 
ous fields  of  research  would  have  been  too  great  a  sacrifice,  had  it  not 
been  for  the  hope  that  I  might  accomplish  something  which  should  be 
of  material  assistance  in  solving  one  of  the  most  difficult  problems  of 
mathematical  instruction,  —  that  of  advanced  calculus. 

EDWIN  BIDWELL  WILSON 
Massachusetts  Ixstitlte  of  Techxologv 


CONTENTS 


INTRODUCTORY  REVIEW 


CHAPTER  I 
REVIEW  OF  FUNDAMENTAL  RULES 

SECTION 

1.  On  differentiation     ....... 

4.  Logarithmic,  exponential,  and  hyperbolic  functions 

0.  Geometric  jiroperties  of  the  derivative 

8.  Derivatives  of  higher  order 

10.  The  indefinite  integral 

13.  Aids  to  integration  .... 

16.  Definite  integrals      .... 


PAGE 

1 
4 

7 
11 
15 
18 
24 


CHAPTEE  II 
REVIEW  OF  FUNDAMENTAL  THEORY 


18.  Xumbers  and  limits 

21.  Theorems  on  limits  and  on  sets  of  points 

23.  Real  functions  of  a  real  variable 

26.  The  derivative  .... 

28.  Summation  and  integration 


33 
37 
40 
45 
50 


PART   I.    DIFFERENTIAL   CALCULUS 


CHAPTER  III 
TAYLOR^S  FORMULxV  AND  ALLIED  TOPICS 

31.  Taylor's  Formula      ........ 

33.  Indeterminate  forms,  infinitesimals,  infinites 

3(5.  Infinitesimal  analysis        ....... 

40.  Some  differential  geometry       ...... 


00 

Gl 
68 
78 


VI 


CONTENTS 


CHAPTER  IV 
PARTIAL  DIFFERENTIATION  ;  EXPLICIT  FUNCTIONS 

SECTION 

43.  Functions  of  two  or  more  variables  ..... 

46.  First  partial  derivatives   ........ 

50.  Derivatives  of  higher  order       ....... 

54.  Taylor's  Formula  and  applications  ...... 

CHAPTER  V 
PARTIAL  DIFFERENTIATION;  IMPLICIT  FUNCTIONS 

56.  The  simplest  case  ;  F(x,i/)  —  0 

59.  More  general  cases  of  implicit  functions 

62.  Functional  determinants  or  Jacobians 

65.  Envelopes  of  curves  and  surfaces 

68.  More  differential  geometry 


PAGE 

87 

93 

102 

112 


117 
122 
129 
135 
143 


CHAPTER  VI 
COMPLEX  NUMBERS  AND  VECTORS 


70.  Operators  and  oj^erations 

71.  Complex  numl)ers     . 

73.  I'unctions  of  a  complex  variable 

75.  Vector  sums  and  products 

77.  Vector  differentiation 


149 
1.53 
157 
163 
170 


PART  II.   DIFFERENTIAL  EQUATIONS 

CHAPTER  VII 
GENERAL  INTRODUCTION  TO  DIFFERENTIAL  EQUATIONS 

81.  Some  geometric  prol)lems  ........ 

83.  Problems  in  mechanics  and  physics  ...... 

85.  Lineal  element  and  differential  equation  ..... 

87.  The  higlier  derivatives  ;  analytic  approximations     .... 

CHAPTER  VIII 
THE  COM.^IOXER  ORDINARY  DIFFERENTIAL  EQUATIONS 

89.  Integration  by  separating  the  variables    ...... 

91.  Integrating  factors  ......... 

95.  Linear  equations  with  constant  coefficients       ..... 

98.  Simultaneous  linear  equations  with  constant  coerticients 


179 
1S4 
191 
197 


203 
207 
214 


CONTENTS  vii 

CHAPTER  IX 
ADDITIONAL  TYPES  OF  ORDINARY  EQUATIONS 

SECTION  PAGE 

100.  Equations  of  the  first  order  and  higlier  degree        ....     228 

102.  Equations  of  higher  order      ........     234 

104.  Linear  differential  equations           .......     240 

107.  The  cylinder  functions  .         .         .         .         .         .         .         .         .247 

CHAPTER  X 
DIFFERENTIAL  EQUATIONS  IN  MORE  THAN  TWO  VARIABLES 

109.  Total  differential  equations    ........  254 

111.  Systems  of  simultaneous  equations         ......  200 

113.  Introduction  to  partial  differential  equations  ....  267 

116.  Types  of  partial  differential  equations    ......  273 


PART  III.   INTEGRAL  CALCULUS 
CHAPTER  XI 

ON  SIMPLE  INTEGRALS 

118.  Integrals  containing  a  parameter  .......  281 

121.  Curvilinear  or  line  integrals  ........  288 

124.  Independency  of  the  path       ........  208 

127.  Some  critical  comments  ........  308 

CHAPTER  XII 
ON  MULTIPLE  INTEGRALS 

129.  Double  sums  and  double  integrals  ......  315 

133.  Triple  integrals  and  change  of  variable  .....  326 

135.  Average  values  and  higher  integrals       ......  332 

137.  Surfaces  and  surface  integrals         .......  338 

CHAPTER  XIII 
ON  INFINITE  INTEGRALS 

140.    Convergence  and  divergence  ........     352 

142.    The  evaluation  of  infinite  integrals         ......     360 

144.    Functions  defined  by  infinite  integrals  ......     368 


viii  CONTENTS 

CHAPTER  XIV 
SPECIAL  FUNCTIONS  DEFINED  BY  INTEGRALS 

SECTION  PAGE 

147.    The  Gamma  and  Beta  fuuctious     .......     378 

1.50.    The  error  fuuctiou  ,.■,......     386 

1.53.    Bessel  functions      ..........     393 


CHAPTER  XV 
THE  CALCULUS  OF  VARIATIONS 

15.5.    The  treatment  of  the  simplest  case         ......     400 

1-57.    Varial:>le  limits  and  constrained  minima  .....     401 

159.    Some  generalizations       .........     409 


PART  IV.    THEORY  OF  FUNCTIONS 
CHAPTER  XVI 
INFINITE   SERIES 

162.    Convergence  or  divergence  of  series        ......     419 

1<J5.    Series  of  functions  .........     430 

108.    Manipulation  of  series    .........     440 

CHAPTER  XVII 

SPECIAL  INFINITE  DEVELOPMENTS 

171.    The  trigonometric  functions  ........     453 

173.    Trigonometric  or  Fourier  series      .......     458 

175.    The  Theta  functions       .........     467 

CHAPTER  XVIII 
FUNCTIONS  OF  A  COMPLEX  VARIABLE 

178.  General  theorems   ..........  476 

ISO.  Characterization  of  some  functions         ......  482 

Us:').  Conformal  representation        ........  490 

l85.  Integrals  and  their  inversion  .......  496 


CONTENTS  ix 

CHAPTER  XIX 

ELLIPTIC  FUNCTIONS  AND  INTEGRALS 

SECTION  PAGE 

187.    Legendre's  integral  I  and  its  inversion  ......  503 

190.    Legendre's  integrals  li  and  III       .......  511 

192.    Weierstrass's  integral  and  its  inversion  ......  517 

CHAPTER  XX 

FUNCTIONS  OF  REAL  VARIABLES 

194.    Partial  differential  equations  of  physics          .....  524 

196.    Harmonic  functions;  general  theorems           .....  530 

198.    Harmonic  functions ;  special  theorems  ......  537 

201.    The  potential  integrals  .........  546 

BOOK  LIST 555 

INDEX 557 


ADVA^^CED  CALCULUS 

INTRODUCTORY  REVIEAV 

CHAPTER  I 

REVIEW  OF  FUNDAMENTAL  RULES 

1.  On  differentiation.  If  the  function  /'(./•)  is  interpreted  as  the 
curve  y  =/(./•),*  the  quotient  of  the  increments  Ay  and  A.r  of  the 
dependent  and  independent  variables  measured  from  (.r^,  y^)  is 

y-//o  _  A//  _  ^fi^l  _  /(r^  +  A.r)-,/Ya-^) 

./•-.Ag       A.r  A./'  Ax  '  ^  ^ 

and  represents  the  sfope  of  the  secant  through  the  points  /^(a'Q,  y^  and 
^''(•%  + '^■'■'  ^0  +  -^'/)  '-*^^  ^^'^  curve.  The  limit  approached  by  the  quo- 
tient \y/ilx  when  P  remains  fixed  and  Aa- =  0  is  the  sIo2jg  of  the 
tangent  to  the  curve  at  the  point  P.    This  limit, 

ii.„^=i.,. /<'-.+ ^;)-/^'»)^/>,),  (2) 

is  called  the  derirative  of  /(x)  for  the  value  x  =  .r^.  As  the  derivative 
may  be  computed  for  different  points  of  the  curve,  it  is  customary  to 
speak  of  the  derivative  as  itself  a  function  of  o:  and  write 

A//  /(.,.  + A,T)-/(.r) 

There  are  numerous  notations  for  the  derivative,  for  instance 

^■'(-■)  =  ^  =  s  =  ^'^= '"''' = *  -  "f=  "'J- 

*  Here  and  tliroiiglioiit  the  work,  where  figures  are  not  given,  the  reader  should  draw 
graphs  to  ilhistrate  the  statements.  Training  in  making  one's  own  iUustrations,  whether 
graphical  or  analytic,  is  of  great  value. 

1 


2  I^iTRODUCTORY  REVIEW 

The  first  five  show  distinctly  that  the  independent  variarble  is  cr,  Avhereas 
the  last  three  do  not  explicitly  indicate  the  variable  and  should  not  be 
used  unless  there  is  no  chance  of  a  misunderstanding. 

2.  The  fundamental  formulas  of  differential  calculus  are  derived 
directly  from  the  application  of  the  definition  (2)  or  (3)  and  from  a 
few  fundamental  propositions  in  limits.    First  may  be  mentioned 


dx  ^  <lf-^  (;/)  _       1      _   1 
dij  dij  df(x)       dy 


(P) 


dx  dx 

D(u  ±  r)  =  Dii  +  Dv,  D{uv)  =  uDv  +  vDu.  (6) 

lu\       rDi(  —  vDv  *  ^^     ^ 

^  (-;  =  — ^. — '      ^  c^")  =  ''^'■"  -'■  (") 

It  may  be  recalled  that  (4),  which  is  the  rule  for  differentiating  a  function  of  a 

function,  follows  from  the  application  of  the  theorem  that  the  limit  of  a  jfroduct  is 

Az       Az  A?/ 

the  product  of  the  limits  to  the  fractional  identity  —  = ~  ;  whence 

Ax      Ay  Ax 

Az        ,.       Az      ..       Au        ,.      ,  Az      ,.       Am 

Inn  — =    Inn  — •    Inn  ^=    limt nm  • — ■, 

Ax  =  0  Ax       A,/:  =  0  Ay    Ax  =  0  Ax       A,v  i  0   Ay    Ax  =  0  Ax 

which  is  equivalent  to  (4).  Similarly,  if  y  =/(x)  and  if  x,  as  the  inverse  function 
of  ?/,  be  written  x=f~'^{y)  from  analogy  with  ?/  =  sinx  and  x  —  sin-'^y.  the 
relation  (5)  follows  from  the  fact  that  Ax/ Ay  and  Ay/ Az  are  reciprocals.  The  next 
three  result  from  the  immediate  application  of  the  theorems  concerning  limits  of 
sum.s,  products,  and  quotients  (§  21).  The  rule  for  differentiating  a  power  is  derived 
in  case  n  is  integral  by  the  application  of  the  binomial  theorem. 

Ay       (x  +  Ax)"— X"  ,       n{n  —  \)         ,.  ,.    x      , 

—  =  ^-^^ =  ?(X"-i  +  — ^ ^x"-2  Ax  +  •  •  •  +    A./-)»-i, 

Ax  Ax  2  !  -r         -I   V     y       5 

and  the  limit  when  Ax  =  0  is  clearlj-  )jx"-i.    The  result  may  be  extended  to  rational 

V  '- 

values  of  the  index  ?i  Iw  writing  n  =  —,  y  =  x'l ,  y''  —  xp  and  by  differentiating 

both  sides  of  the  equation  and  reducing.  To  prove  that  (7)  still  holds  when  ji  is 
irrational,  it  would  be  necessary  to  have  a  icorkahle  definition  of  irrational  numbers 
and  to  develop  tlie  properties  of  sucli  luimbers  in  greater  detail  than  seems  wise  at 
this  point.  The  fornmla  is  therefore  assumed  in  accordance  with  the  princijAe  of 
permanence  of  form  (§178).  just  as  formulas  like  a"'a"  =  a"'  +  "  of  the  theory  of 
exponents,  which  may  readily  be  proved  for  rational  bases  and  exponents,  are 
assumed  without  proof  to  hold  also  for  irrational  bases  and  exponents.  See,  how- 
ever, §§  18-25  and  the  exercises  thereunder. 

*  It  is  frecpicntly  better  to  regard  the  quotient  as  tlie  proiluet  x  ■  r-i  and  apply  (6). 
t  For  wlien  A-r  =  0,  then  Ay  =  0  or  \ij/\:c  could  not  approach  a  limit. 


FUXDAMEXTAL  KULES  3 

3.  Second  may  be  mentioned  the  formulas  for  the  derivatives  of  the 
trigonometric  and  the  inverse  trigonometric  functions. 

D  sin  .T  =  cos  0I-,  D  cos  x  =  —  sin  x,  (8) 

or                 D  sin  x  —  sin  (x-  +  i  tt),  D  cos  x  =  cos  (x  +  J  tt),  (8') 

D  tana;  =  sec-.r,  D  cot  x  =  —  csc^x,  (9) 

D  sec  X  =  sec  a-  tan  x,  D  esc  ,/■  =  —  esc  x  cot  r,  (10) 

D  vers  a'  =  sin  x,     where      vers  x  =1  —  cos  x  =  2  sin'-^  |^  rr,  (11) 

^    .      1  ±1  f  +  in  (luadrants      I,  lY,         ,^  ^. 

i)sm-..  =  -/=__,  |_^,   ^      ,,  jj^j^^;       (12) 

r  —  in  quadrants      I,    II,         ,^  o\ 
i+"  "         III,  IV,        ^^'^) 

/;cot-ia-  =  -^,,         (14) 

r  +  in  quadrants      I,  III,         ,^  w, 

I-"    "    11,  iv;   (1^) 

r  —  in  quadrants     I,  III,         ,^  ,.. 
I  +   "  "  II,  IV,        ^      -* 

r  +  in  quadrants      I,    II,         ,^  -^ 
V2J^^''  i-"  "         III,  IV.        ^-^'^ 

It  may  be  recalled  that  to  differentiate  sinx  the  definition  is  applied.    Then 

Asinx      sin  (x  +  Ax)  —  sinx      sin  Ax  1— cos  Ax   . 

= = cos  X sni  X. 

Ax  Ax  Ax  Ax 

It  now  is  merely  a  question  of  evaluating  the  two  limits  which  thus  arise,  namely, 

sin  Ax                         1— cos  Ax  .^.. 

lim  and      lun (18) 

Axio     Ax  A.<  =  0  Ax 

From  the  properties  of  the  circle  it  follows  that  these  are  respectively  1  and  0. 
Hence  the  derivative  of  sinx  is  cosx.  The  derivative  of  cosx  may  be  found  in 
like  manner  or  from  the  identity  cos  x  =  sin  (^  tt  —  x) .  The  results  for  all  the  other 
trigonometric  functions  are  derived  by  expressing  the  functions  in  terms  of  sin  x 
and  cos  x.  And  to  treat  the  inverse  functions,  it  is  sufficient  to  recall  the  general 
method  in  (5).    Thus 

if         i/  =  sin-ix,  then         siiiy  =  x. 


Differentiate  both  sides  of  the  latter  equation  and  note  that  cos?/  =  ±  Vl  —  sin-y 
=  ±  Vl  —  X-  and  the  result  for  D  sin-^x  is  innnediate.  To  ascertain  which  sign  to 
use  with  the  radical,  it  is  .sufficient  to  note  that  ±  Vl  —  x-  is  cosy,  which  is  po.sitive 
when  the  angle  ?/  =  sin-ix  is  in  quadrants  I  and  IV,  negative  in  II  and  III. 
Similarly  for  the  other  inverse  functions. 


4  INTRODUCTORY  REVIEW 

EXERCISES  * 

1.  Carry  through  the  derivation  of  (7)  when  n  —p/q,  and  review  the  proofs  of 
typical  fonnulas  selected  from  the  list  (5)-(17).    Note  that  the  formulas  are  often      ^ 
given  as  DxU"  =  nu'^-^D^u,  D^sinu  =  con uD^u,  •  •  •,  and  may  be  derived  in  this 
form  directly  from  the  definition  (3). 

2.  Derive  the  two  limits  necessary  for  the  differentiation  of  sinx. 

3.  Draw  graphs  of  the  inverse  trigonometric  functions  and  label  the  portions 
of  the  curves  which  correspond  to  quadrants  I,  II,  III,  IV.  Verify  the  sign  in 
(12)-(17)  from  the  slope  of  the  curves. 

4.  Find  Z>tanx  and  I>cotx  by  applying  the  definition  (3)  directly. 

5.  Find  D  sinx  by  the  identity  sin  u  —  sin  v  =  2  cos sin -■ 

2  2 

«  u  —  V 

6.  Find  D  tan-^x  by  the  identity  tan-i  u  —  tan-i  v  =  tan-i and  (3). 

1  +  uv 

7.  Differentiate  the  following  expressions  : 

(a)  CSC  2  X  —  cot  2  x,     (^)  I  tan^x  —  tan  x  +  x,     (7)  x  cos-i  x  —  Vl  —  x'-, 


(5)  .sec-i  —  .        (e)  sin-i ,  (f)  x  Va-  —  x-  +  a-  .sin-i  -  , 

Vl  -  X-  Vl  +  x-  " 


2rtx 


(ij)  a  ver.s-i  -  —  Vz ((X  —  x-,       {0)  cot-i  — ^ 

tt  X"  —  u-  II 

What  trigonometric  identities  are  suggested  by  the  answers  f(jr  the  following: 

(«)  .sec^x,  {5)     1 .  (^)  rr-^'  (^)  0- 

VI     -      X-  1      +      ^^- 

8.  In  B.  O.  Peirce's  "  Short  Table  of  Integrals"  (revi.sed  edition)  differentiate  the 
right-hand  members  to  confirm  the  formulas  :  Xos.  31,  45-47,  01-U7,  125,  127-128, 
131-135,  1(J1-1G3,  214-210,  220,  200-2GU,  2U4-298,  300,  380-381,  38(i-3'.i4. 

9.  If  X  is  measured  in  degrees,  what  is  iisinx  '? 

4.  The  logarithmic,  exponential,  and  hyperbolic  functions.    The 

next  set  of  formulas  to  be  cited  are 

1  -1  loir,/'  ,^^^ 

]j  log^.r  =  - ,  D  log„.r  =  — ^ ,  (19) 

De'  =  e^,  7>/*  =  rr'loy,.'^t  (20) 

It  may  be  recalled  that  the  procedure  for  differentiating  the  logarithm  is 

X 

Alo(j„x      log„(x  +  Ax)  —  l()ir„x        1   ,        x  +  Ax      1,        /,       AxXaj 

log„  — =  -  log,,    1  +       ' 


Ax  Ax  Ax  X  X  \  X 

*  The  student  should  keep  ou  file  his  solutions  of  at  least  the  important  exercises: 
many  subsequent  exercises  and  consideraljle  portions  of  the  text  depend  on  previous 
exercises. 

t  As  is  customary,  the  subscript  e  will  IxTeafter  be  omitted  and  tlie  .symbol  log  will 
denote  the  logaritluu  to  tlie  base  e;  any  base  other  than  f  must  Ije  si)ecially  designated 
as  such.  This  observation  is  particularly  neces.sary  with  reference  to  the  connnou  Ijase 
10  used  in  computation. 


FUNDAMENTAL  RULES 

If  now  x/Ax  be  set  equal  to  A,  the  problem  becomes  that  of  evaluating 


H'^j) 


6  =  2.71828..-,*  logioe  =  0.4342y-4...;  (21) 


and  hence  if  e  be  chosen  as  the  base  of  the  system,  D  log  x  takes  the  simple  form 
1/x.  The  exponential  functions  e^  and  a^  may  be  regarded  as  the  inverse  functions 
of  logx  and  logaX  in  deducing  (21).  Further  it  should  be  noted  that  it  is  frequently 
useful  to  take  the  logarithm  of  an  expression  before  differentiating.  This  is  known 
as  logarithmic  differentiation  and  is  used  for  products  and  complicated  powers  and 
roots.    Thus 

if  y  =  X'',  then         log?/  =  x  logx, 

and         -y'  —  1  +  logx  or  i/  =  x''{l  +  higx). 

It  is  the  expression  y'/y  which  is  called  the  logarithmic  derivative  of  y.  An  especially 
noteworthy  property  of  the  function  y  =  Ce^  is  that  the  function  and  its  derivative 
are  equal,  y'  =  y  ;  and  more  generally  the  function  y  =  Ce^'-*  in  proportional  to  ita 
derivative,  y'  =  ky. 

5.   The  lu/perhollc  fiinrtlinui  are  the  hyperbolic  sine  and  cosine, 

sinli  ./•  =  '''''' J  ' ,  cosh  X  =  '^'^\''     ;  (22) 

and  the  related  functions  tanh./-,  coth  .r,  sech./',  cs<;h.r,  derived  from 
them  by  the  same  ratios  as  those  by  which  the  corresponding  trigono- 
metric functions  are  derived  from  sinx  and  cos.;c.  From  tliese  defini- 
tions in  terms  of  exponentials  follow  the  formulas  : 

cosh"-;/"  —  sinlr./'  =  1,  tanlr./'  -f-  seclr./'  =  1,  (23) 

sinh  (./•  ±  >/)  =  sinh  x  cosh  y  ±  cosh  x  sinh  //,  (24) 

cosh  (./'  ±  i/')  =  cosli  X  cosh  //  ±  sinh  x  sinh  //,  (25) 

,  X              fcosh.r-l-l             .   ,   .'■        ,       fcosha-  —  1      ,^^, 
cosh  -  =  +  ^ ^ ,         snih  -  =  ±  ^ ^ ,  (26) 

X*  sinh  ./•  =  cosh  .r,  y^*  cosh  ,/•  =  sinh  r,  (27) 

D  tanh  ./■  =  sech-^:*',  D  coth  x  =  —  csch'-.r,  (28) 

D  sech  ,/•  =  —  sech  ./•  tanh  x,     D  csch  ,/•  =  —  csch  ./•  coth  x.    (29) 

The  inverse  functions  are  expressible  in  terms  of  logarithms.    Thus 

e-K  —  l 
y  =  sinh  ^A",  X  =  sinh  ii  =  -— , ? 

e-y  —  2  xe"  —  1=0,  e^=:  X  ±  ■\fx-  + 1. 

*  The  treatment  af  this  limit  is  far  from  complete  in  the  majority  of  texts.  Reference 
for  a  careful  presentation  may,  however,  be  made  to  (iranville's  '"  Caleulus,"  pp.  .'31-54, 
and  Osgood's  "Calculus,"  pp.  78-82.    See  also  Ex.  1,  (/3),  in  §  IGo  below. 


6  I^sTRODUCTORY  REVIEW 

Here  only  the  positive  sign  is  available,  for  e^  is  never  negative.  Hence 

sinh-^  X  =  log  {x  -f-  Vx^  + 1),              any  x,  (30) 

cosli~"^a:!  =  log(.x' ±  V^:'^  — l),              x  >  1,  (31) 

tanh-^a;  =  -  log  ^-— ^ ,                        x^  <1,  (32) 


1        a'+l 
coth-i  x  =  ~  log  -— -  ,  x^  >  1,  (33) 

sech-i  x  =  log  (  -  ±  ^—  -  1  j  ,  a;  <  1,  (34) 


any  a-, 

X  >\, 

x'Kl, 

x'  >  1, 

X   <1, 

csch-i  X  =  log  (  -  +  J-  +  1  j  ,  any  x,  (35) 

J>sinli-^a;=      ,  >       i;  cosli-^a;  =  — -^=,  (36) 

Va--^  +  l  Vx-^-l  ^     ^ 

D  tanh-ia-  =  ^ r,  =  />  coth-ia^  =  :; ;,  (37) 

1  —  X'  1  —  a-  ^      ' 

i)  sech-i  X  =  = — — ,     D  csch-i  a-  =  ■ ~  •  (38) 

a'  VI  —  ./■-  a-  V 1  +  a;" 


EXERCISES 

1.  Show  by  logarithmic  differentiation  that 

1)  {uvw  ...)  =  ^  +  -  +  —  +  ...    {uvw  ■ .  .), 
\uvw  I 

and  hence  derive  the  rule  :  To  differentiate  a  product  differentiate  eacli  factor 
alone  and  add  all  the  results  thus  obtained. 

2.  Sketcli  the  graphs  of  tlie  hyperbolic  functions,  interpret  tlie  graphs  as  those 
of  the  inverse  functions,  and  verify  tlie  range  of  values  assigned  to  x  in  (80)-(3.5). 

3.  Prove  sundry  of  formulas  (23)-(20)  from  the  definitions  (22). 

4.  Prove  sundry  of  (30)-(;->8),  checking  the  signs  with  care.  In  cases  where 
double  signs  remain,  .state  wlien  each  applies.  Note  that  in  (31)  and  (34)  the 
double  sign  may  be  j)laced  before  tlie  log  for  the  reason  that  the  two  expressions 
are  reciprocals. 

5.  Derive  a  fonnula  for  sinli);  ±  sinlir  by  applying  (24)  ;  find  a  foruuda  for 
tanh  \  X  analogous  to  the  trigonometric  fonnula  tan  Ix  =  sinx/(l  +  c(.isx). 

6.  The  (judermannian.    The  function  4>  =  gd.c.  defined  by  the  relations 

sinh  X  =  tan  (f>,     <p  =  gd  x  =  tan-i  sinli  .c,     —  i,  tt  <  0  <  +  \  tt, 
is  called  the  gudermanniau  of  x.    Prove  the  set  of  fornndas  : 
cosh  X  =  sec  0,     taidi  x  =  sin  0,     cscli  x  =  cot  0,     etc.  ; 
IJ  gd  X  =  secli  X,     X  =  gd-i  (p  =  log  tan  (',  0  +  J-  tt),     IJ  gd-i  4>  =  sec  (f>. 

7.  Substitute  the  functions  of  (p  in  Kx.  (>  for  their  liyperbolic  equivalents  in 
(23),  (26),  (27),  and  reduce  to  simple  known  trigonometric  formulas. 


FUNDAMENTAL  RULES  7 

8.  Differentiate  the  following  expressions  : 

(a)  (X  +  l)-(x  +  2)-3(x  +  3)-S         {13)  x'o=-,  (7)  log,.(x  +  1), 

(5)  X  +  logcos(x  —  1  tt),  (e)  2tan-ie^,         (j")  x  — tanlix, 

(ij)  X  tanh-ix  +  i  log(l  -  x-),  ((9)  — ^ '-■ 

m-  +  u- 

9.  Check  sundry  formulas  of  Peirce's  "Table,""  pp.  1-01,  81-82. 

6.  Geometric  properties  of  the  derivative.  As  the  quotient  (1)  and 
its  limit  (2)  give  tlie  slope  of  a  secant  and  of  the  tangent,  it  appears 
from  graphical  considerations  that  when  the  derivative  is  positive  the 
function  is  increasing  with  x,  but  decreasing  when  the  derivative  is 
negative.*  Hence  to  determine  the  regions  in  irhieh  a  f  11  net  ion  is  in- 
creasing or  decreasing,  one  may  Jind  the  deriratire  and  deterntine  the 
regions  in  which   it  is  positive  or  negatirc. 

One  must,  however,  be  careful  not  to  apply  this  rule  too  blindly  ;  for  in  so 
simple  a  case  as/(x)  =  logx  it  is  seen  that/'(x)  =  1/x  is  positive  when  x  >  0  and 
negative  when  x  <  0,  and  yet  log  x  has  no  graph  when  x  <  0  and  is  not  considered 
as  decreasing.  Thus  the  formal  derivative  may  be  real  wlien  the  function  is  not 
real,  and  it  is  therefore  best  to  make  a  rough  sketch  of  the  function  to  cfjrroborate 
the  evidence  furnished  by  the  examination  of /'(x). 

If  oc^  is  a  value  of  a-  such  that  immediately  t  upon  one  side  of  .r  =  x^ 
the  function  /"(.'')  is  increasing  whereas  immediately  upon  the  other 
side  it  is  decreasing,  the  ordinate  ij^=f(.r^)  will  be  a  maximum  or 
minimum  or  ,f  (•'')  ^^^^■'-  become  positively  or  negatively  infinite  at  :r^. 
If  the  case  where  /(■/')  becomes  infinite  be  ruled  out,  one  may  say  that 
tit e  function  vill  hare  a  inininniin  or  niaxiiiuini  at  .r^  accurding  as  tlie 
deriratire  changes  from  negotice  to  jiositli'c  or  from  j>ositire  to  ncgatlre 
irlien  x,  moving  in  tlie  positive  direction,  passes  through  the  value  x^. 
Hence  the  tisual  rule  for  determining  maxima  and  minima,  is  to  find 
the  roots  of  f'(x)  =  0. 

This  rule,  again,  must  not  be  applied  blindly.  For  first, /'(x)  may  vanish  where 
there  is  no  maxinuim  or  mininuim  as  in  the  case  y  =  x'^  at  x  =  0  where  the  deriva- 
tive does  not  change  sign;  or  second, /'(x)  maj'  change  sign  by  becoming  inlinite 
as  in  the  case  ?/  =  x^  at  x  =  0  where  the  curve  has  a  vertical  cusp,  i)oint  down,  and 
a  minimum  ;  or  third,  the  function /(x)  may  be  restricted  to  a  given  range  of  values 
a  ^  X  ^  6  for  x  and  then  the  values /(a)  and/(6)  of  the  function  at  the  ends  of  the 
interval  will  in  general  be  maxima  or  minima  without  implying  that  the  deriva- 
tive vanish.  Thus  although  the  derivative  is  highly  usefvd  in  determining  maxima 
and  minima,  it  should  not  be  trusted  to  the  complete  exclusion  of  the  corroborative 
evidence  furnished  by  a  rough  sketch  of  the  curve  y  =/(x). 

*  The  construction  of  illustrative  figures  is  again  left  to  the  reader. 

t  The  word  "  innnediately  "  is  necessary  because  the  niaxinui  or  ininima  may  be 
merely  relative;  iu  the  case  of  several  maxima  and  nnninia  in  an  interval,  sonu'  of 
the  maxima  may  actually  be  less  than  some  of  the  minima. 


INTEODUCTOPvY  EEVIEW 


7.  The  derivative  may  be  used  to  express  the  equations  of  the  tangent 
and  normal,  the  values  of  the  subtangent  and  subnon/ial,  and  so  on. 

Equation  of  tangent,      y-y^  =  y[  (^  -  ^o)'  (39) 

Equation  of  normal,   (//  —  y^)  i/[  -|-  {.c  —  a-^)  =  0,  (40) 

TM  =  subtangent  =  yjij^,     MX  =  subnormal  =  y^ij[,  (41) 

6*7'  =  u'-intercept  of  tangent  =  j-^  —  y^/y[,  etc.  (42) 

The  derivation  of  tliese  results  is  sufficiently  evi- 
dent from  the  figure.  It  may  be  noted  that  the 
subtangent,  subnormal,  etc.,  are  numerical  values 
for  a  given  point  of  the  curve  but  may  be  regarded 
as  functions  of  x  like  the  derivative. 
In  geometrical  and  jjliysical  problems  it  is  frequently  necessary  to 
apply  the  definition  of  the  derivative  to  finding  the  derivative  of  an 
unknown  function.  For  instance  if  A  denote  the 
area  under  a  curve  and  measured  from  a  fixed 
ordinate  to  a  variable  ordinate,  A  is  surely  a  func- 
tion A{x)  of  the  abscissa  x  of  the  variable  ordinate. 
If  the  curve  is  rising,  as  in  the  figure,  then  q  ,/  ^^^ 

MPQ'M'  <  A,l  <  MQP'M',   or  i/\x  <  A.I  <  (//  +  Ay)  Ax-. 
Divide  by  A./j  and  take  tlie  limit  when  Ax  =  0.    There  results 


Hence 


lim  //  ^  Jim  

A  J-  =  0  Aa:  =  0   A.t' 

,.       AJ 

lim  

aj  =  o  a.'' 


lim  (//  + A//). 

AxiO 
d.l 


(43) 


liolle^s  Theorem  and  the  Theorem  if  the  Mrnn  are  two  important 
theorems  on  derivatives  which  will  be  treated  in  the  next  chapter  but 
may  here  V)e  stated  as  evident  from  their  geometric  interpretation. 
li()lh'"s    Theorem   states   that  :    Jf  "  functuni   has  a  di-rlmtu'e  at  rrt-ri/ 


Y 

^ 

J 

A 

^' 

B 

O 

f 

Fig.  1 


Fi( 


Fi( 


l>nint  of  (in  Interval  and  f  the  fum-tioti  ranishrs  at  the  ends  if  tin-  In- 
ti'rral^  tlwn  tlwre  is  at  least  one  puird  u-ithin  the  intercaJ  at  adiii-h  tin' 
drrieatiri'  rmiishrs.  This  is  illustrated  in  Fig.  1,  in  which  there  ai'e 
two  such  }i()ints.    The  Theorem  of  the  Mean  states  that:   If  a  funitiun 


FUNDA]\[EXTAL  RULES  9 

has  a  derivative  at  each  point  of  an  intcn-al,  there  is  at  least  one  itoint 
in  the  interval  such  that  the  tawjent  to  the  carce  i/=f(:r')  is  jjarallel  to 
the  chord  of  the  intcrral.  Tliis  is  illustrated  in  Fig.  2  in  which  there 
is  only  one  such  point. 

Again  care  must  be  exercised.  In  Fig.  3  the  function  vanislies  at  A  and  B  but 
there  is  no  point  at  which  the  slope  of  the  tangent  is  zero.  This  is  not  an  excep- 
tion or  contradiction  to  Rolle's  Theorem  for  the  reason  that  the  function  does  not 
satisfy  the  conditions  of  the  theorem.  In  fact  at  the  point  P,  although  there  is  a 
tangent  to  the  curve,  there  is  no  derivative  ;  the  quotient  (1)  formed  for  the  point  P 
becomes  negatively  infinite  as  Ac  =  0  from  one  side,  positively  infinite  as  Ax  =  0 
from  the  other  side,  and  therefore  does  not  approach  a  definite  limit  as  is  required 
in  the  definition  of  a  derivative.  The  hypothesis  of  the  theorem  is  not  satisfied  and 
there  is  no  reason  that  the  conclusion  should  hold. 

EXERCISES 

1.  Determine  the  regions  in  which  the  following  functions  are  increasing  or 
decreasing,  sketch  the  graphs,  and  find  the  maxima  and  minima  : 

{a)  i  x^-x-  +  2.  (^)   (x  +  l)^{x-  of.  (y)  log {x"  -  4), 


(3)   {x-2)Vx-l.        ( e )  _  (X  +  2) V 1 2  -  x\         (f)  x^  +  ax  +  b. 


2.  The  ellipse  is  r  =  Vx' +  y- =  e{d  +  x)  referred  to  an  origin  at  the  focus. 
Fin<l  the  maxima  and  minima  of  the  focal  radius  r,  and  state  why  D^r  =  0  does 
not  give  the  solutions  while  D^r  =  0  does  [the  polar  form  of  the  ellipse  being 
r  =  k{l  —  e  cos0)-i]. 

3.  Take  the  ellipse  as  x-/(i.-  +  y-/b-  =  1  and  discuss  the  maxima  and  minima  of 
the  central  radius  r  =Vx-  +  (/-.  Why  does  iJji-  =  0  give  half  the  result  when  r  is 
expressed  as  a  function  of  x.  and  why  will  JJ^r  =  0  give  the  whole  result  when 
X  =  a  cosX,  y  =  ?;sinX  and  the  ellipse  is  thus  expressed  in  terms  of  the  eccentric 
angle  '? 

4.  If  2/  =  I'{x)  is  a  polynomial  in  x  such  that  the  equation  7'  {x)  =  0  has  nudtiple 
roots,  show  that  P'{x)  =  0  for  each  multiple  root.  What  more  complete  rL-lationsliip 
can  be  stated  and  proved  ? 

5.  Show  that  the  triple  relation  27  b~  -|-  4  a^  S  0  determines  completely  the  nature 
of  the  roots  of  x^  +  ax  +  b  =  0,  and  state  what  corresponds  to  each  possibility. 

6.  Define  the  angle  0  beta-een  two  interserting  curves.    .Siiow  that 

tan  0  =  [/'(,/:„)  -  ;/'(./;,)]  -^  [1  +r{x,)y'{x,)] 
if  y  =f(x)  and  y  =  g  ix)  cut  at  the  point  [x^^.  y^). 

7.  Find  the  subnormal  and  subtangent  of  the  three  curves 

(a)  y-  =  Ajjx,  (p)  X-  -  4py,  (y)  x-  +  y"  =  a-. 

8.  The  pedal  curve.  The  locus  of  the  foot  of  the  perpendicular  dropped  from 
a  fixed  point  to  a  vaiiable  tangent  of  a  given  curve  is  called  the  pedal  of  the  given 
curve  with  respect  to  the  given  point.  Show  that  if  the  fixed  point  is  the  origin, 
the  pedal  of  y  =f{x)  may  be  obtained  b}'  eliminating  Xy,  y^,  y'o  from  the  equations 

y  -yn  =  y',  (■'•  -  a'u),    yy'j  +  x  =  o.    y,,  =  /{x„),    y;^  =  f'(x^). 


10  IXTKODUCTORY  REVIEW 

Find  the  pedal  {a)  of  the  hyperbola  with  respect  to  the  center  and  (/3)  of  the 
parabola  with  respect  to  the  vertex  and  (7)  the  focus.  Show  (5)  that  the  pedal  of 
the  parabola  with  respect  to  any  point  is  a  cubic. 

9.  If  the  curve  y  =/(x)  be  revolved  about  the  z-axis  and  if  V(x)  denote  the 
volume  of  revolution  thus  generated  when  measured  from  a  fixed  plane  perpen- 
dicular to  the  axis  out  to  a  variable  plane  perpendicular  to  the  axis,  show  that 

i^xV  =  -n-r  • 

10.  ^lore  generally  if  A  {x)  denote  the  area  of  the  section  cut  from  a  solid  by 
a  plane  perpendicular  to  the  x-axi.s,  show  that  DxV  =  A  (x). 

11.  If  .4  {(p)  denote  the  sectorial  area  of  a  plane  curve  r  =f{<p)  and  be  measured 
from  a  fixed  radius  to  a  variable  radius,  show  that  D^A  =  |  /•-. 

12.  If  p,  h.  p  are  tlie  density,  height,  pressure  in  a  vertical  column  of  air,  show 
that  dp/dh  =—  p-    If  p  =  kp,  show  p  —  Ce-^''. 

13.  Draw  a  graph  to  illustrate  an  apparent  exception  to  the  Theorem  of  the 
Mean  analogous  to  the  apparent  exception  to  Rolle"s  Theorem,  and  discuss. 

14.  Show  that  the  analytic  statement  of  the  Theorem  of  the  Mean  forf(x)  is 
that  a  value  x  =  ^  intermediate  to  a  and  h  may  be  found  such  that 

f{b)  -f(a)=  r  (I)  (6  -  «),  a  <  ^  <  h. 

15.  Show  that  the  semiaxis  of  an  ellipse  is  a  mean  proportional  between  the 
z-intercept  of  the  tangent  and  the  ab.sci.ssa  of  the  point  of  contact. 

16.  Find  the  values  of  the  length  of  the  tangent  (a)  from  the  point  of  tangency 
to  the  z-axis,  (/3)  to  the  y-axis.  (7)  the  total  length  intercepted  between  the  axes. 
Consider  the  same  problems  for  the  normal  (figure  on  page  8). 

17.  Find    the    angle    of    intersection    of     (a)  y-  =  2  nix    and    x-  +  y-  =  a-. 

,„.     o       ,              1                 8r/^                        X-                y-  for    i)<K<h 

(3)  x"  —  4  ay   and   y  = ,     (7) =  1  ,  ,  ^     ^ 

18.  A  constant  length  is  laid  off  along  the  normal  to  a  parabf)la.    Find  the  locus. 

19.  The  length  of  the  tangent  to  x^  +  y'^  =  a^  intercepted  by  the  axes  is  con.stant. 

20.  The  triangle  formed  by  the  asymptotes  and  any  tangent  t(j  a  hyperbola  has 
constant  area. 


21.  Find  the  length  FT  of  the  tangent  to  x  =  Vc-  —  y-  +  c  sech-^  (2/A). 

22.  Find  tlie  greate.st  right  cylinder  inscribed  in  a  given  right  cone. 

23.  Find  the  cylinder  of  greatest  lateral  .surface  in.scribed  in  a  sphere. 

24.  From  a  given  circular  sheet  of  metal  cut  out  a  sector  that  will  form  a  cone 
(without  base)  of  maximum  volume. 

25.  Jnin  two  points  ^-1.  B  in  the  same  side  of  a  line  to  a  point  P  uf  the  line  in 
such  a  way  that  the  distance  PA  +  PB  shall  be  least. 

26.  Obtain  the  formula  for  the  distance  from  a  point  to  a  line  as  the  minimum 
distance. 

27.  Test  for  maximum  or  minimum,  {a)  \i  f(x)  vanishes  at  the  ends  of  an  inter- 
val and  is  positive  within  the  interval  and  if  f'{x)  =  0  has  only  one  root  in  the 
interval,  that  root  indicates  a  maximum.  Prove  this  by  Rollers  Theorem.  Apply 
it  in  Exs.  22-24.  (^)  If  fix)  becomes  indefinitely  great  at  the  ends  of  an  interval 
and/'(x)  =  0  has  only  one  root  in  the  interval,  that  root  indicates  a  uunimum. 


FUNDAMENTAL  RULES  11 

Prove  by  RoUe's  Theorem,  and  apply  in  Exs.  25-26.  These  rules  or  various  modi- 
fications of  them  geierally  suffice  in  practical  problems  to  distinguish  between 
maxima  and  minimc  without  examining  either  the  changes  in  sign  of  the  first 
derivative  or  the  f/gn  of  the  second  derivative  ;  for  generally  there  is  only  one 
root  oif'(x)  =  Oin  the  region  considered. 

28.  Show  chat  x~^  sinx  from  x  =  0  to  x  =  ^tt  steadily  decreases  from  1  to  2/7r. 

1  iz- 

29.  If  J<z  <  1,  show  (a)  0  <x- log(l  +  x)  < -z2,  (/3) -^ — <  x  -  log(l  +  «). 

2  1  +  X 

1  i  X- 

30.  if  0  >  X  >  —  1,  show  that  -  x^  <  x  —  log  (1  +  x)  <    " 


2  °'  1  +  x 

P.  Derivatives  of  higher  order.  The  derivative  of  the  derivative 
regarded  as  itself  a  function  of  ./•)  is  the  second  derivative,  and  so  on 
,0  the  nth.  derivative.    Customary  notations  are : 

/•"w,/-w,  ■■■,y^-'W;  g>g' ■■■>£'••■• 

The  Tith  derivative  of  the  sum  or  difference  is  the  sum  or  difference  of 
the  nth.  derivatives.  Eor  the  nth  derivative  of  the  product  there  is  a 
special  formula  known  as  LeUjnlz's  Theorem.    It  is 

D''(i(v)  =  D"u  ■  v  +  nir-^ uDc  +  ^^      ~      ir--uD'i- -\ ^uD^u.   (44) 

This  result  may  be  written  in  symbolic  form  as 

Leibniz's  Theorem     D"-{uv)  =  {Du  +  Di-y,  (44') 

where  it  is  to  be  understood  that  in  expanding  (/)?/  -f  7)/;)"  the  term 
(Duy  is  to  be  replaced  by  L^ti  and  (Day  by  B'^ii  =  v.  In  other  words 
the  powers  refer  to  repeated  differentiations. 

A  proof  of  (44)  by  induction  will  be  found  in  §  27.    The  following  proof  is 
interesting  on  account  of  its  ingenuity.    Note  first  that  from 

D  (UI-)  =  uDv  +  vDu,     D"  (uv)  =  D  (uDv)  +  D  (i-Du), 

and  so  on,  it  appears  that  D-  {uv)  consists  of  a  sum  of  terms,  in  each  of  which  there 
are  two  differentiations,  with  numerical  coefficients  independent  of  w  and  v.  In  like 
manner  it  is  clear  that 

I)"  (Mr)  =  CqD'>u  ■  V  +  C^D^-^uBv  +  •  •  •  +  C„  _i  7>wZ»«  -i  i-  +  C„uD''v 

is  a  sum  of  terms,  in  each  of  which  there  are  ri  differentiations,  with  coefficients  C 
independent  of  u  and  v.    To  determine  the  C"s  any  suitable  functions  u  and  v,  say, 

may  be  substituted.    If  the  substitution  be  made  and  e'-^+"^^'  be  canceled, 
e-(i-n).,-7)«(„,)  =  (1  +  ,f)n  =  C,  +  C,a  +  .  .  .  +  C„_ia"-i  +  C„a-, 
and  hence  the  C"s  are  the  coefficients  in  the  binomial  expansion  of  (1  +  «)". 


12  INTRODUCTORY  REVIE^ 

function  may  be 
Formula  (4)  for  the  derivative  of  a  function      ,^        crenerally 
extended  to  higher  derivatives  by  repeated  applic  '       .    i         ^^f  f\\ 
any  desired  chunrje  of  variable  may  he  made  hy  tht^  ^  functions 

and  (o).  For  if  x  and  y  be  expressed  in  terms  of  ,  .i  ^leriva- 
of  new  variables  u  and  r,  it  is  alwaj's  possible  to  obta^  ^.gggJQj^ 
fives  D^y,  D'^y,  ■  •  ■  in  terms  of  Z),,?-,  D^r,  ■  ■  ■,  and  thus  auj.^^.gggjoj^ 
F(x,  y,  y',  y",  ■  •  •)  may  be  changed  into  an  equivalent  .^g  ^\^q 
^(ji,  r,  v',  r",  •  •  •)  in  the  new  variables.  In  each  ease  that  a.p  /n 
transformations  should  be  carried  out  by  repeated  application 
and  (o)  rather  than  by  substitution  in  any  general  formulas. 

The  following  typical  cases  are  illustrative  of  the  method  of  change  of  variab 
Suppose  only  the  dependent  variable  y  is  to  be  changed  to  z  defined  as?/=/(2).   Tb 

d-y  _  d  /dy\  _  d  /dz  dy\  _  d-z  dy  dz  /  d  dy\ 

dz^      dx  \dx/      dx  \dx  dzj      dx-  dz  dx  \dx  dz) 

_  d-z  dy      dz  /  d  dy  dz\  _  d-z  dy  A^~\"  d-y 

,                      dx-  dz      dx  \dz  dz  dx)      dx"  dz  \dx)    dz- 

As  the  derivatives  of  y  =f{z)  are  known,  the  derivative  d-y/dx-  has  been  expr 
in  terms  of  z  and  derivatives  of  z  with  respect  to  x.  The  third  derivative  won 
found  by  repeating  the  process.  If  the  X'roblem  were  to  change  the  indepe 
variable  x  to  2,  defined  by  x  =/(2), 

dy  _  dy  dz  _  dy  (dx\-^  ^^"v  _  '^  \<Ml  ('^'\^ 

dx      dz  dx      dz  \dzj  dx-      dx\_dz  \dzj     J 

d-y  _  d-y  dz  /rZ.r\-i     dy  /dj-\--  dz  d-x  _  Vd-y  dx      d-x  dy~\      A 
dx-       dz:-  dx\dz)         dz\dz/       dx  dz-       tdz-  dz       dz-  dzj      \< 


The  change  is  thus  made  as  far  as  derivatives  of  the  second  order  are  conceri 
the  change  of  both  dependent  and  independent  variables  was  to  be  made,  th 
would  be  similar.  Particularly  useful  changes  are  to  find  the  derivatives  of 
when  y  and  x  are  expressed  parametrically  as  functions  of  t,  or  when  both  ; 
pressed  in  terms  of  new  variables  r,  <p  as  x  —  r  cos  (p.  y  =  r  sin  (p.  For  these 
see  the  exercises. 

9.   The  mnrtn-ify  of  n  nirrc  y^f{.r^  is  giv(Mi  by  the  ta])le : 

if    /"(■''f)  >  ^,  tlie  curve  is  concave  up  at  ,r  =  .^v,, 

if    f"(.r^)  <  0,  the  curve  is  concave  down  at  :r  =  r^, 

if    /"(■'',)  =  0,  an  inflection  point  at  ,/•  =  .'■..  (?) 

Hence  the  criterion  for  distiyir/uisliiyir/  hrtireen  muxiiini  and  ruinirini 
if    f{x^  =  0    and  /'"(.''J  >  0,  a  minimum  at  ./•  =  .r^,, 

if    /''(./•.)  =  0    and  .fV-'V,)  <  0,  a  maximum  at  ,r  =  ./■_,, 

if    /'(,/;^)  =  0    and  /"(./'j  =  0,  neither  max.  nor  min.  (? 


FUNDAMENTAL  RULES  13 

The  question  points  are  necessary  in  the  third  line  because  the  state- 
ments are  not  always  true  unless /'"(a-^j)  ^  0  (see  Ex.  7  under  §  39). 

It  may  be  recalled  that  the  reason  that  the  curve  is  concave  up  in  case/"(XQ)  >  0 
is  because  the  derivative  /'(x)  is  then  an  increa.sing  function  in  the  neighborhood 
of  x  —  x^;  whereas  if /"(x^)  <  0,  the  derivative /'(x)  is  a  decreasing  function  and 
the  curve  is  convex  up.  It  should  be  noted  that  concave  up  is  not  the  same  as 
concave  toward  the  x-axis,  except  when  the  curve  is  below  the  axis.  AVith  regard 
to  the  use  of  the  second  derivative  as  a  criterion  for  distinguishing  between  maxima 
and  minima,  it  should  be  stated  that  in  practical  examples  the  criterion  is  of  rela- 
tively small  value.  It  is  usually  shorter  to  discuss  the  change  of  sign  of /'(x)  directly, 
—  and  indeed  in  most  cases  either  a  rough  graph  of /(x)  or  the  physical  conditions 
of  the  problem  which  calls  for  the  determination  of  a  maximum  or  minimum  will 
immediately  serve  to  distinguish  between  them  (see  Ex.  27  above). 

The  second  derivative  is  fundamental  in  dynamics.  By  definition  the 
ai-emge  velocity  v  of  a  particle  is  the  ratio  of  the  space  traversed  to  the 
time  consumed,  v  =  s/t.  The  actual  velocitij  v  at  any  time  is  the  limit 
of  this  ratio  when  the  interval  of  time  is  diminished  and  approaches 
zero  as  its  limit.    Thus 

r  =  — ^     and     V  =  lim  -—  =  -r  •  ('45") 

M  At  =  0^t       clt  ^      ' 

In  like  manner  if  a  particle  describes  a  straight  line,  say  the  cr-axis,  the 
(ivHrnge  accelerntion  f  is  the  ratio  of  the  increment  of  velocity  to  the 
increment  of  time,  and  the  actual  acceleration  f  ^t  any  time  is  the  limit 
of  this  ratio  as  A^  =  0.    Thus 

/=—     and    /=hm —  =  —  =  —•  (46) 

By  Newton's  Second  Law  of  JlTotion,  the  force  actinfj  on  the  particle  is 
equal  to  the  rate  of  cliancje  of  momentum,  with  the  time,  momentum 
being  defined  as  the  product  of  the  mass  and  velocity.    Thus 

iljun-^  (h-  d-r 

I<  =  — ; — -  =  m  —r  =  ri>  f  =  m  ——;  >  (4 < ) 

dt  dt         '  dt-  ^     ' 

where  it  has  been  assumed  in  differentiating  that  the  mass  is  constant, 
as  is  usually  the  case.  Hence  (47)  appears  as  the  fundamental  e(pia- 
tion  for  rectilinear  motion  (see  also  §§  79,  84).    It  may  be  noted  that 

dr       d  [1       A      dT  ,,_. 

where  7'=  i  n)r~  denotes  by  definition  the  kinetic  enercjy  of  the  particle. 
For  comments  see  Ex.  6  following. 


14  INTRODUCTORY  REVIEW 

EXERCISES 

1.  State  and  prove  the  extension  of  Leibniz's  Theorem  to  products  of  tliree  oi 
more  factors.    Write  out  the  square  and  cube  of  a  trinomial. 

2.  Write,  by  Leibniz's  Tlieorem,  the  second  and  third  derivatives : 

(a)  e^sinx,  (/3)  cosh x  cos x,  (7)  x-e^logx. 

3.  Write  the  nth  derivatives  of  tlie  following  functions,  of  which  the  last  three 
should  first  be  simplified  by  division  or  separation  into  partial  fractions. 


(a)  Vx  +  1,  (^)  \og{ax  +  b),  (7)  (x2  +  1)  (x  +  l)-3, 

(5)  cosax,  (e)  e^sinx,  (f)  (1  —  x)/(l  +  x), 


iv) 


1  ...   x^  +  X  4- 1  ,  .    /ax  +  IV 


x^-l 


X  —  1  \c 


4.  If  y  and  x  are  each  functions  of  i,  show  that 
dx  d"y      dy  d^x 
d'^y       dt  df^       dt  df^       x'y"  —  y'x'' 


dx2                 /dx\3                       X'3 

\dt) 

d^y      x'  {x'y'"  —  y'x'")  —  3  x"  (x'y" 

—  y'x") 

dx^                                  x'^ 

5.  Find  the  inflection  points  of  the  curve  x  =  4  0  —  2  sin  ^,  ?/  =  4  —  2  cos  </>. 

6.  Prove  (47').    Hence  infer  that  the  force  which  is  the  time-derivative  of  the 
momentum  mv  by  (47)  is  also  the  space-derivative  of  the  kinetic  energy. 

7.  If  A  denote  the  area  under  a  curve,  as  in  (4.3),  find  dA/d9  for  the  curves 

{a)  y  =  a{l—  cos 0),  x  =  a{9  —  sin 6),       (fi)  x  —  a  cos 0,  y  =  b  .sin 0. 

8.  Make  the  indicated  change  of  variable  in  the  following  equations: 

,    ,   d-y         2x     dy  V  r,  .  a         ^^?/ 

(a)   — ^  H -\ =  0,     X  =  tan  z.  Ans.    —^  +  y  =:  0. 

dx^       1  +  x^  dx      (1  4-  x-y-  dzr 


X  =  sm  II. 

Ans.    ----1-1  =  0. 
du" 

9.  Transformation  to  polar  coordinates.  Suppose  that  x  =  7*  cos  0,  ?/  =  r  sin  </>.  Then 

dx       dr  .  dy       dr    . 

—  =  —  cos  (p  —  r  am  4>,         —  =  —  sm  ^  +  r  cos  </>, 

d(f>      dif)  d(p      d(p 

,      ,.,        ,     .     ,.  ,,.    ,dy  rPy      r^^  +  2{D^r)"^  -  rl)^r 

and  so  on  for  higher  derivatives.    I  md  —  and  = 

dx  dx^       (cos  <p  B^r  —  r  sin  4>)^ 

10.  Generalize  formula  (5)  for  the  differentiation  of  an  inver.se  function.    Find 
d"-x/dy"  and  d'^'x/dy^.    Note  that  tliese  may  also  be  found  from  Ex.  4. 

11.  A  point   describes  a  circle  with  constant  speed.    Find  the  velocity  and 
acceleration  of  the  projection  of  the  point  on  any  fixed  diameter. 

12.  Prove  — ^  =  2 nv  +  4v*  (  —  )     —  v ( —  )       if  x  =  -•  y  —  uv. 

dx-  \du/  du^  \du/  v 


FUNDAMENTAL  KULES  15 

10.  The  indefinite  integral.  To  integrate  a  function  /(p-)  is  to  faid 
a  function  F(3i-)  tlte  derivative  of  vlilch  is  f(x).  The  integral  F(r)  is 
not  uniquely  determined  by  the  integrand  ,/'(•') '  ^^^'  ^^^y  ^^^^  funcjtions 
which  differ  merely  by  an  additive  constant  have  the  same  derivative. 
In  giving  formulas  for  integration  the  constant  may  be  omitted  and 
understood ;  but  in  applications  of  integration  to  actual  problems  it 
should  always  be  inserted  and  must  usually  be  determined  to  fit  the 
requirements  of  special  conditions  imposed  upon  the  problem  and 
known  as  the   initial  conditions. 

It  must  not  be  thought  that  the  constant  of  integration  always  appears  added  to  tlie 
function  F(x).    It  may  be  combined  with  F(x)  so  as  to  be  somewhat  disguised.  Thus 

logx,       h.igx  +  C,       logCx,       log(x/(7) 

are  all  integrals  of  1/x,  and  all  except  the  first  have  the  constant  of  integration  C, 
although  only  in  the  second  does  it  appear  as  formally  additive.  To  illustrate  the 
determination  of  the  constant  by  initial  conditions,  consider  the  problem  of  finding 
the  area  under  the  curve  y  =  cosx.    By  (43) 

DjcA  =  y  =  cos  X     and  hence     A  —  sin  x  +  C. 

If  the  area  is  to  be  measured  from  the  ordinate  x  =  0,  then  A  =  0  when  x  =  0,  and 
by  direct  substitution  it  is  seen  that  C  =  0.  Hence  A  =  sin  x.  But  if  the  area  be 
measured  from  x=— |7r,  then  ^=0  when  x  =  — Itt  and  C  =1.  Hence  ^  =  l  +  sinx. 
In  fact  the  area  under  a  curve  is  not  definite  until  the  ordinate  from  which  it  is 
measured  is  specified,  and  the  constant  is  needed  to  allow  the  integral  to  fit  this 
initial  condition. 


11.   The  fundamental  formulas  of  intcGfration  are  as  follows: 


(48) 


I  e^  =  e%                                           I  a-'  =  «71og  a,  (49) 

J.i„.,.  =  -cos.,                         Jcos.  =  sin..  (60) 

I  tan  X  =  —  log  cos  .'■,                    /  cot  .r  =  log  sin  x,  (51) 

I  sec'a-  =  tan  x,                               I  csc'a;  =  —  cot  x,  (o2) 

I  tan  X  sec  x  =  sec  x,                     j  cot  x  esc  x  =  —  esc  x,  (53) 

with  formulas  similar  to  (o0)-(5.3)  for  the  hyperbolic  functions.  Alrio 

/- :,  —  tan~^a-  or  —  cot"\'',     I  :, ;  =  tanh"^r  or  coth~'.r,  (54) 
1  +  ^-                                              J   1-x- 


1(3  i:n^troductory  review 

/I       .  r   ±  1        . 

— ;=  =  siii'^T  01"  —  COS  '?■      I  —  =  +  sinh'^r,  (55) 

Vl-^--  J   Vl  +  ar  ^ 

f     1  r    ±  1 

I  —  =  sec~^j'or  —  CSC  \/',    |  —  =p  sech'^j-,         (56) 

r      ,^^        =  +  COsh-^r,  r ^^—  =  qP  CSch-l.r,  (57) 

J    Vx'-l  J    .rVl  +  a- 

/      ,^  =  vers"',/-,         /  sec./'  =  gd-^x  =  logtaiW  -  +':;)•  (''58) 

For  the  integrals  expressed  in  terms  of  the  inverse  hyperbolic  functions,  the 
logarithmic  equivalents  are  sometimes  preferable.  This  is  not  the  case,  however, 
in  the  many  instances  in  which  the  problem  calls  for  immediate  solution  with 

regard  to  x.    Thus  if  y  =  I  (1  +  x-)~  %  =  sinh-i  x  +  C,  then  x  =  sinh  (y  —  C),  and  tiie 

.solution  is  effected  and  may  be  translated  into  exponentials.  This  is  not  so  easily 
accomplished  from  the  form  y  =  log  (x  +  v  1  +  x-)  +  C.  For  this  reason  and 
because  the  inverse  hyperbolic  functions  are  briefer  and  offer  striking  analogies 
with  the  inverse  trigonometric  functions,  it  has  been  thought  better  to  use  them 
in  the  text  and  allow  the  reader  to  make  the  neces.sary  substitutions  from  the  table 
(.30)-(35)  in  case  the  logarithmic  form  is  desired. 

12.  In  addition  to  these  special  integrals,  vvliich  are  consequences 
of  the  corresponding  formulas  for  differentiation,  there  are  the  general 
rules  of  integration  which  arise  from  (4)  and  (6). 

j^>f  +  r  -  ir)  =    j   u^    j    r-    j   >r,  (60) 

vr=    (  in-' -\-    I  ii'r.  ((il) 

Of  these  rules  the  second  needs  no  cnnnnent  and  the  tliird  will  be  treated  hitcr. 
Especial  attention  should  be  given  to  the  tirst.  For  instance  suppose  it  were  re- 
(luired  to  integrate  2  logx/x.    This  does  not  fall  under  any  of  the  given  types  ;  but 

2     ^^     _d  (log  x)2  d  logx  _  dz  dy 
X     °  d  logx        dx  dy  dx 

Here  (logx)-  takes  tlie  place  of  z  and  logx  takes  the  place  of  y.  Tlie  integral  is 
therefore  (logx)-  as  may  be  verified  by  differentiation.  In  general,  it  may  be 
po.ssible  to  .see  that  a  given  integrand  is  separable  into  two  factf)rs.  of  which  one 
is  integrable  when  considered  as  a  finiction  of  some  function  of  x.  while  the  other 
is  the  derivative  of  that  function.    Then  (-^O)  applies.    Other  examples  are  : 


I  t^'"''  cosx,         rtan-ix/(l  +  x"),         fx-  sin  (x' 


FUNDAMENTAL  RULES  17 

In  the  first,  z  =  C'  is  integrable  and  as  ?/  =  sin  a;,  y'  =  cosx  ;  in  tlie  second,  z  =  ?/  is 
integrable  and  as  y  =  tan-ix,  2/'=  (1  +  x-)-i  ;  in  tlie  third  z  =  siny  is  integrable 
and  as  ?/  =  a;^,  y'  =  3x~.    Tlie  results  are 

c^'n^         I  (tan-i  a;)2,         —  1  cos  (x'') . 

Tliis  nietliod  of  integration  at  sight  covers  sucli  a  large  percentage  of  the  cases 
that  arise  in  geometry  and  physics  that  it  must  be  thoroughly  mastered.* 

EXERCISES 

1.  Verify  the  fundamental  integrals  (48)-(58)  and  give  the  hyperbolic  analogues 
of  (o0)-(53). 

2.  Tabulate  tlie  integrals  here  expressed  in  terms  of  inverse  hyperbolic  func- 
tions by  means  of  the  corresponding  logarithmic  equivalents. 

3.  "Write  the  integrals  of  the  following  integrands  at  sight: 

(/3)  cot{ax  +  b),  (7)  tanhSx, 


{a} 

SlUrtX, 

(5) 

1 

<l~  +  X- 

iv) 

1 

xlogx 

(x) 

x^vox-  +  b, 

{") 

(x-1  -  1)^ 

X- 

ip) 

C6l  +  '*'"^C0SX, 

Vx-  —  a"  V2  ox  —  x^ 

1 

X-  X-  +  a- 

(X)  tanxsec"x,  (/ic)   cot  x  log  sin  x, 

,  ,  taidi-^x  ,   ,   2  +  loijx 

(0) ;;-,  (tt)  —  , 

1  —  X"  X 

.sinx  1 

(<,)       _^^,  (r) 


Vcosx  Vl  — x-sin-ix 

4.  Integrate  after  making  appropriate  changes  such  as  sin-x  =  ^  —  J  cos  2x 
or  sec-x  =  1  4- tan"x,  division  of  denominator  into  numerator,  resolution  of  the 
product  of  trigonometric  functions  into  a  sum,  completing  the  square,  and  so  on. 

(a)  COS-2X,  (/3)  sin^x.  (7)  tau*x, 

;c2  +3x4-2.')'  ^^'    X4-2'  ^^'      versx 

/    \  •'^  +  •'>  tm  (■-''  +  t"  ,  .  1 


4 x2  -  5 X  +  1  e^'^  +  1  V2 ax  +  x^ 

(k)  sin  5x  cos  2x  +  ],  (X)  sinln?(X  sinluix,  {/x)  cosx  cos  2x  cos  3x, 

, cj-  _|_  ,;  rm—l 

(v)  sec^x  tanx  —  V2x,  (o)  ^ -,  (tt) 


x"  +  ax  +  b  {(ix">  +  b)i' 

*  The  use  of  dilTerentials  (§  .'55)  is  perliaps  more  familiar  than  tlio  use  of  derivatives. 

J    dx  J    (l>i  dx  J    di/ 


Then 


I  ~  leg  X  d.i-  =  I  -  log  X  d  log  X  =  (log  x)-. 


The  use  of  this  notation  is  left  optional  with  the  reader;  it  has  some  advantages  and 
some  disadvantages.  The  essential  thing  is  to  keep  clearly  in  mind  the  fact  tliat  the 
prol)lem  is  to  be  inspected  with  a  view  to  detecting  the  function  which  will  differentiate 
into  the  given  integrand. 


18  INTRODUCTOKY  EEVIEW 

5.  How  are  the  following  types  integrated  ? 

(a)  sin"'x  co.s"x,  m  or  n  odd,  or  m  and  n  even, 

(^)  tan"x  or  cot"x  when  n  is  an  integer, 

(7)  sec"x  or  cscx  when  n  is  even, 

(5)  tan'"x  seC'x  or  cot'"x  csc"x,  n  even. 

6.  Explain  the  alternative  foniis  in  (54)-(56)  with  all  detail  possible. 

7.  Find  {a)  the  area  under  the  parabola  y-  =  A^px  from  x  =  0  to  x  =  a  ;  also 
(P)  the  corresponding  volume  of  revolution.  Find  (7)  the  total  volume  of  an  ellij)- 
soid  of  revolution  (see  Ex.  9,  p.  10). 

8.  Show  that  the  area  under  y  =  sin  mz  sin  nx  or  y  =  cos  itix  cos  nx  from  x  =  0 
to  X  =  TT  is  zero  if  m  and  n  are  unequal  integers  but  4  ir  if  they  are  ecjual. 

9.  Find  the  sectorial  area  of  r  =  a  tan  0  between  the  radii  0  =  0  and  <p  =  ^tt. 

10.  Find  the  area  of  the  (a)  lemniscate  ?•-  =  «-  cos20  and  (^3)  cardioid  r=l  — cos0. 

11.  By  Ex.  10,  p.  10,  find  the  volumes  of  these  solids.  Be  careful  to  choose  the 
parallel  planes  so  that  A  (x)  may  be  found  easily. 

{a)  The  part  cut  off  from  a  right  circular  cylinder  Ijy  a  plane  through  a  diameter 
of  one  base  and  tangent  to  the  other.  Aim.  2/3  tt  of  the  whole  volume. 

(j3)  How  much  is  cut  off  from  a  right  circular  cylimler  by  a  plane  tangent  to  its 
lower  base  and  inclined  at  an  angle  ff  to  the  plane  of  the  base  '? 

(7)  A  circle  of  radius  b  <  a  is  revolved,  about  a  line  in  its  j^lane  at  a  distance  a 
from  its  center,  to  generate  a  ring.    Tlie  volume  of  the  ring  is  2Tr'-ah'-. 

(5)  The  axes  of  two  equal  cylinders  of  revolution  of  radius  r  intersect  at  right 
angles.    The  volume  common  to  the  cylinders  is  10 /""'/S. 

12.  If  the  cross  section  of  a  solid  is  A(x)  =  a,yC^  +  a^x-  +  a.,x  +  a.^.  a  cubic  in  x, 
the  volume  of  the  solid  between  two  i)arallel  planes  is  1  h  (Ji  +  i  M  +  B')  where  h 
is  the  altitude  and  B  and  B'  are  the  bases  and  M  is  the  middle  section. 


13.   Show  that    f =  tan- 

J  1  +  X-  1  —  ex 


.1  -g  +  c 


13.  Aids  to  integration.  The  majority  of  eases  of  integration  wltich 
arise  in  simple  a})})Iieations  of  ealeulus  may  ])e  treated  by  tlu^  method 
of  §  12.  Of  the  remaining  eases  a  large  number  eannot  Ije  integrated 
at  all  in  terms  of  the  functions  whieli  have   boon  treated  u})  to  this 

point.    Thus  it  is  impossiljle  to  ex})ress  /  =  in  terms 

J     V(l-./'-)(l-r/V-) 

of  elementary  functions.  One  of  the  ehief  reasons  for  introducing  a 
variety  of  new  functions  in  higher  analysis  is  to  liave  means  for  effeet- 
ing  the  integrations  called  for  by  important  applications.  Tlic  dis- 
cussion of  this  matter  cannot  be  taken  u[)  here.  The  problem  of 
integration  from  an  elementary  point  of  view  calls  for  the  tal)ula- 
tion    of    some    tleviees    which    Avill    accomplish    the    integration    for   a 


rUXDAMEXTAL  RULES  19 

wide  variety  of  integrands  integrable  in  terms  of  elementary  functions. 
The  devices  wliich  will  be  treated  are : 

Integration  by  parts,  Resolution  into  partial  fractions, 

Various  substitutions.         Reference  to  tables  of  integrals. 
Integration  by  parts  is  an  application  of  (61)  when  written  as 

I  ur'  =  uv  —  i  u'v.  (61') 

That  is,  it  may  happen  that  tlie  integrand  can  be  written  as  the  prodnct  uv'  of  two 
factors,  where  v'  is  integrable  and  where  u'v  is  also  integrable.  Then  uv'  is  integrable. 
For  instance,  logx  is  not  integrated  by  the  fundamental  formulas  ;  but 

I  logx  =  I  logx  •  1  =  X  logx  —  I  x/x  =  X  lugx  —  X. 

Here  log  x  is  taken  as  u  and  1  as  v',  so  that  i-  is  x.  u'  is  1/x,  and  u'v  =  1  is  immedi- 
ately integrable.  This  method  applies  to  the  inverse  trigonometric  and  hyperbolic 
functions.    Another  example  is 

/  X  sin  X  =  —  X  cos  x  +  /  cos x  =  sin  x  —  x  cos x. 

Here  if  x  =  w  and  sinx  =  v'.  both  v'  and  u'v  =—  cosx  are  integrable.  If  the  choice 
sin  x  —  u  and  x  =  v'  had  been  made,  v'  would  have  been  integrable  but  u'v  =  l  x-  cosx 
would  have  been  less  simple  to  integrate  than  the  original  integrand.  Hence  in 
apph'ing  integration  by  parts  it  is  necessary  to  look  ahead  far  enough  to  see  that 
both  v'  and  u'v  are  integrable,  or  at  any  rate  that  v'  is  integrable  and  the  integral 
of  u'v  is  simpler  than  the  original  integral.* 

Frequentlj' integration  by  parts  has  to  be  applied  .several  times  in  succession.  Thus 

I  x'-e'  =  x-f  —  I  2xe-''  if  u  —  x-,  v'  =  e^', 

=  x-c-''  —  2    xe'  —  I  e'-  ii  u  =  x.    v'  =  e^, 

=  x-t-''  —  2  xe-''  +  ■2e^. 

Sometimes  it  may  be  applied  in  such  a  way  as  to  lead  back  to  the  given  integral 
and  thus  afford  an  equation  from  which  that  integral  can  be  obtained  by  solution. 
For  example, 

I  e-'"  cosx  =  e^  cosx  +  I  e''  sinx  if  u  =  cosx,  v'  =  e-"", 

=  e^(cosx  +  sinx)  —  |  e-'cosx. 
Hence  |  e^cosx  =  l  e''(cosx  +  sinx). 

*  The  nietliod  of  differentials  may  again  be  introduced  if  desired. 


20  INTRODUCTORY  REVIEW 

14.  For  the  integration  of  a  rational  fraction  f  {x)  /  F  (x)  where /and  i^  are  poly- 
nomials in  X,  the  fraction  is  first  resolved  into  partial  fractions.  This  is  accom- 
plished as  follows.  First  if  /  is  not  of  lower  degree  than  F,  divide  F  into  /  until  the 
remainder  is  of  lower  degree  than  F.  The  fraction  f/F  is  thus  resolved  into  the 
sum  of  a  polynomial  (the  quotient)  and  a  fraction  (the  remainder  divided  by  F) 
of  which  the  numerator  is  of  lower  degree  than  the  denominator.  As  the  polyno- 
mial is  integrable,  it  is  merely  necessary  to  consider  fractions  f/F  where  /  is  of 
lower  degree  than  F.  Next  it  is  a  fundamental  theorem  of  algebra  that  a  poly- 
nomial F  may  be  resolved  into  linear  and  quadratic  factors 

F  (x)  =  A:  (x  —  a)"  (x  —  h)P  (x  —  c)v  .  .  .  {x"  +  mx  +  71)1^  {x"  +  px  +  qy  ■  ■  ■ , 

where  a,  6,  c,  •  ■  •  are  the  real  I'oots  of  the  equation  F{x)=  0  and  are  of  the  respec- 
tive multiplicities  a,  /3,  7,  •  •  •,  and  where  the  quadratic  factors  when  set  (Mjual  to 
zero  give  the  pairs  of  conjugate  imaginary  roots  of  F  =  0,  the  multiplicities  of  the 
imaginary  roots  being  /x,  v,  ■  ■  ■ .  It  is  then  a  further  theorem  of  algebra  that  the 
fraction //F  may  be  written  as 

fix)         A,             A.,  Aa             B,                        Bb 

F{x)      X  —  a      (x  —  a)-  (x  —  u)''      x  —  h                (x  —  h)li 

M^x  +  N^  J/.X  +  X,      +  .  .  .  +      -"^^M-g  +  A>           ^  _  _ 

X-  4-  mx  +  n  (x-  +  mx  +  )t)-                 (x-  +  mx  +  n)t^ 

where  there  is  for  each  irreducible  factor  of  F  a  term  corresponding  to  the  highest 
power  to  which  that  factor  occurs  in  F  and  also  a  term  corresponding  to  every 
lesser  power.  The  coefficients  A^  B,  •  •  •,  M,  N,  ■  ■  ■  may  be  obtained  by  clearing 
of  fractions  and  equating  coefficients  of  like  powers  of  x,  and  solving  the  e(]uations  ; 
or  they  may  be  obtained  by  clearing  of  fractions,  substituting  for  x  as  many  dif- 
ferent values  as  the  degree  of  F,  and  solving  the  resulting  equations. 

When  f/F  has  thus  been  resolved  into  partial  fractions,  the  problem  has  been 
reduced  to  the  integration  of  each  fraction,  and  this  does  not  present  serious 
difficulty.  The  following  two  examples  will  illustrate  tlie  metliod  of  resolution 
into  partial  fractions  and  of  integration.    Let  it  be  recjuired  to  integrate 

r  x-  +  1  ,     /-      2  x3  +  (5 

(  :; and ; r- 

J  X  (X  -  1)  (X  -  2)  (X-  +  X  +  1)  J  (X  -  1)2  (X  -  Hf 

The  first  fraction  is  expansible  into  partial  fractions  in  the  form 

X-  4-  1  A  B  C  J)x  +  F 

=  -  + 7  + ^  + 


X  (x  —  1)  (x  —  2)  (x-  +  X  -I-  I)       X       X  —  1      X  —  2      X-  4-  X  +  1 

Hence        x-  -1-  1  =  ^  (x  -  1)  (x  -  2)  (x'-  -1-  x  +  1)  +  Bx  (x  -  2)  (x-  -1-  x  -|-  1) 
-t-  Cx  (X  -  1)  (x"-  -^  X  4-  1)  +  {Dx  4-  E)  X  (X  -  ])  (x  -  2). 

Rather  than  multiply  out  and  equate  coefficients,  let  0,  1.  2.  —  1.  —  2  be  substi- 
tuted.   Then 

1  =  2.1,     2  =  -  3  /.',     5  =  14  C,     D-E  =  1/21,     K  -•ll)='i /7, 

r  X-  +  1  —  C  ^         C       ~  C        ^  r       ix  +  5 

J  x(x-l)(x-2)(x-4-^+l)  ~  -'  2  X  ~  J  ;J  (X  ^Y)      J  ]  4  (X  -  2)      J  21  (./,■-  +  .'•  +  !) 

1  2                         o                           2  2  L'x+1 
=  ~logx-     log(x  — 1)+     Aoiiix—  2) lo<r(x2  4-x  4-1) tau~- 

2  ^  S  14     ■  21  7v'o  VS 


FUNDAMENTAL  KULES  21 

In  the  second  case  the  form  to  be  assumed  for  the  expansion  is 

2  x«  +  6  A  B  C  D  E 

+  - Z7:,  +  ^, ^  +  z ^  + 


(X  -  1)-  (X  -  3)'^      X  -  1       (X  -  1)2       (X  -  3)       (x  -  3)-       (X  -  3)3 
2x3  +  6  =  ^(x-l)(x-3)3  +  B(x-3)3  +  C'(x-l)2(x-3)2 
+  i>  (X  -  1)-  (x  -  3)  +  £■  (X  -  1)2. 
The  substitution  of  1,  3,  0,  2,  4  gives  the  equations 

8=-8  7>,         60  =  4£',         9.4  + 3C'-Z)  + 12  =  0, 
.4-C'  +  Z>  +  0  =  0,         ^  +  3C  +  3Z)  =  0. 
The  solutions  are  —  9/4,  —  1,  +  9/4,  —  3/2,  15,  and  the  integral  becomes 


/ 


2x3  +  0  9,      ,        ,,  1  9,      , 

-  -  log  (X  -  1)  + h  -  log  (x  -  3) 


(X  -  1)-  (X  -  3)3  4        '  X  -  1      4 

3  15 

+ 


2  (X  -  3)      2  (X  -  3)- 

The  importance  of  the  fact  that  the  method  of  partial  fractions  shows  that  any 
rational  fraction  may  he  integrated  and,  moreover,  that  the  integral  may  at  most  con- 
sist of  a  rational  part  plus  the  logarithm  of  a  rational  fraction  plus  the  inverse 
tangent  of  a  rational  fraction  should  not  be  overlooked.  Taken  with  the  method 
of  substitution  it  establishes  very  wide  categories  of  integrands  wliich  are  inte- 
grate in  terms  of  elementary  functions,  and  effects  their  integration  even  though 
by  a  somewhat  laborious  method. 

15.  The  metJiod  of  substitution  depends  on  the  identity 

r/(x)=  r/[^(2/)]^  if         x  =  <t>{y),  (59') 

Jx  Jy  dy 

which  is  allied  to  (59).    To  show  that  the  integral  on  the  right  with  respect  to  y 

is  the  integral  of  /(x)  with  respect  to  x  it  is  merely  necessary  to  show  that  its 

derivative  with  respect  to  x  is  /(x).    By  definition  of  integration, 

'^      r    j-r       /XT  C^X  .r       ,     XT  <^^x 

dyJf/  dy  dy 

and  y  f  n<P{y)]j  =n<P{y)]~-'^=n<p{!/)] 

dxJy  dy  dy    dx 

by  (4).  The  identity  is  therefore  proved.  The  method  of  integration  by  substitu- 
tion is  in  fact  seen  to  be  merely  such  a  systematizatiou  of  the  method  based  on 
(59)  and  set  forth  in  §  12  as  will  make  it  practicable  for  more  complicated  problems. 
Again,  differentials  may  be  used  if  preferred. 

Let  E  denote  a  rational  function.    To  effect  the  integratifju  of 

I  sinx  Zi  (sin-x,  cosx),        let     cosx  =  ?/,         then     l—It(\—y-^y); 

I  cosx  l?(cos'-x,  sinx),        let     sin  x  =  ?/,         then    j  R{1  —  y'-,  y) ; 

fn(^^'^]=  fj:{tiiux),    let     tanx  =  2/,         then     fj^^^"*- 
J        Vcosx/       -^  '^y  I  - 


I  7?  (sinx,  cosx),  let     tan-  =  t/,         then     I  ^'(-^ 


+  2/- 

2?/       1—2/- 


-+y-    l+y-/l  +  y- 

The  last  substitution  renders  any  rational  function  of  sin  x  and  cos  x  rational  in 
the  variable  y ;  it  sliould  not  be  used,  however,  if  the  previous  ones  are  applicable 
—  't  is  almost  certain  to  give  a  more  difficult  final  rational  fraction  to  integrate. 


jR  (x,  V«-  +  X-)    J, 


Cr  (x,  Vx-  -  a^ 


22  mTRODUCTORY  REVIEW 

A  large  number  of  geometric  problems  give  integrands  which  are  rational  in  x 
and  in  some  one  of  the  radicals  Va-  +  x-,  Va-  —  x^,  Vx-  —  a-.  These  may  be  con- 
verted into  trigonometric  or  hyperbolic  integrands  by  the  following  substitutions: 

I  R  (x,  Vu-  —  X-)        X  =  a  sin  y,         (  R  (a  sin  y,  a  cos  y)  a  cos  y ; 

I  tan  y,         j  R  {a  tan  y,  a  sec  y)  a  sec-  y 
'J  y 

\  X  =  a  sinh  ?/,       |  R  (a  sinh  ?/,  a  cosh  ?/)  a  cosh  y  ; 
I  X  =  a  sec  2/,         |  fi  (rx  .sec  y,  a  tan  ?/)  a  sec  ?/  tan  ?/ 
I  X  =  a  cosh  ?/,       I  i?  («  cosii  y,  a  sinh  y)  a  sinh  ?/. 

It  frequently  turns  out  that  the  integrals  on  the  right  are  easily  obtained  by 
methods  already  given ;  otherwise  they  can  be  treated  by  the  substitutions  above. 

In  addition  to  these  substitutions  there  are  a  large  number  of  others  which  are 
applied  under  specific  conditions.  Many  of  them  will  be  found  among  the  exer- 
cises. Moreover,  it  frequently  happens  that  an  integrand,  which  does  not  come 
under  any  of  the  standard  types  for  which  substitutions  are  indicated,  is  none  the 
less  integrable  by  some  substitution  which  the  form  of  the  integrand  will  suggest. 

Tables  of  integrals,  giving  the  integrals  of  a  large  number  of  integrands,  have 
been  constructed  by  using  various  methods  of  integration.  B.  O.  Peirce's  "  Short 
Table  of  Integrals  "  may  be  cited.  If  the  particular  integrand  which  is  desired  does 
not  occur  in  the  Table,  it  may  be  possible  to  devise  some  substitution  which  will 
reduce  it  to  a  tabulated  form.  In  the  Table  are  also  given  a  large  number  of 
reduction  fornuilas  (for  the  most  part  deduced  by  means  of  integration  by  parts) 
which  accomplish  the  successive  simplification  of  integrands  which  could  perhaps 
be  treated  by  other  methods,  but  only  with  an  excessive  amount  of  labor.  Several 
of  these  reduction  fornuilas  are  cited  among  the  exercises.  Although  the  Table  is 
useful  in  performing  integrations  and  indeed  makes  it  to  a  large  extent  unneces- 
sary to  learn  the  various  methods  of  integration,  the  exercises  immediately  below, 
which  are  constructed  for  the  purpose  of  illustrating  methods  of  integration,  should 
be  done  without  the  aid  of  a  Table. 

EXERCISES 

1.  Integrate  the  following  by  parts  : 

(a)   /xcoshx,  (^)   I  tan-ix,  (7)    Cx"'\ogx, 

^    ^J       x^  ^  ^J  il  +  xf  ''^J  xix^-a^)^ 

2.  If  P(x)  is  a  polynomial  and  P'{x),  P"{x),  ■  ■  ■  its  derivatives,  show 

(a)   fP{x)  e"-'-  =  -  e"-^  \p  (x)  -  ^-  P'(x)  +  -  P"{x) 1 , 

•^  a       L  "  a-  J 

(iS )    fl'  (X)  cos  ax  =  ^  sin  cu:  \p  (x)  -  ^  P'\x)  +  -  P"(j\ 1 

J  "  L  "'  "'  J 

+  -  cos  nx  \^  T"(x)  -  -  P"'{x)  +  \  P-{x) 1, 

((  \_(i  a'  r(^  J 

and  (7)  derive  a  similar  result  for  tlic  integrand  P [x)  sinr/x. 


(a)   I  f'^' m\hx  = 


o-  +  h- 
&"•*■  (b  sin  bx  +  a  cos bx) 


FUNDAMENTAL   KULES  23 

3.  By  successive  integration  by  parts  and  subsequent  solution,  sliow 
t"-''  (a  sin  bx  —  b  cos  bx) 

r"  Mil  I/.C  = 

{(3)   fe<'-cn^bx  , 

^  a-  +  b- 

(7)   /  xc--'-  cosx  =  2Ve--»'[5x(sina;  +  2  cosx)  —  4  sinx  —  3  cosx]. 

4.  Pr(jve  by  integration  by  parts  the  reduction  fornudas 

,    ^    r  ■  sin"'+ix  cos"-ix       n  —  lr. 

(a)    I  sni"'xcos"x  = 1 I  sin'"x  cos"--x, 

,,,,    r,  tan"'-ix  sec"x  »( —  1        r 

(fi)   I  tan"'xsec"x  = |  tan" --x  sec" x, 

^  m  +  ji  —  1  ?/i  +  )i  —  1  J 

(7)  r^^- — ^ — r — - — +i^n-s)  f — ' — - 

J  (x^  +  a^y      2  {n  -  I)  a- 1  (x'-=  +  a-)''-i      ^  '  J  (x-  +  a-)"  -1 

(5)   f-^^= "^^ m+lr^c^ 

J  {\0'j:x)»  ()i- l)(logx)«-i       n-lJ    (iogx)"-i 

5.  Integrate  by  decomposition  into  partial  fractions  : 

(5)    r '" ,  ^4x^-3x4-1^  in   f—' 

^      J  (x  +  2)-{x  +  1)  -J        2 x^  +  x3  ^   '  J  x{l  +  x-f 

6.  Integrate  by  trigonometric  or  liyperbolic  substitutii)n  : 

(a)  jx'a-  -  X-.  {(i)  JVx^  -  u\  (7)  fVa-  +  x-, 

(5)/--L^,  (e)/^^^^,  (r)/*'^^. 

-'0/-X-)i  J  X  ^  xi 

7.  Find  the  areas  of  tliese  curves  and  tlieir  volumes  of  revulution  : 

(«)  xt + (/? = «t,       (/J)  «v- = "-.'•^  -  j«.    (7)  ('^y+(0'=i. 

8.  Integrate  by  converting  to  a  rational  algebraic  fraction  : 

r  sinSx  ,„^    T  cosSx  ,   .     r  sin2x 

U^)    I r-'       (■3)    h-. ; ; '         (7)    I  ; ; 

J  a-  C(,>s-  X  +  b-  sin-  x  J  (/-  cos-  x  -f-  '>  sin-  x  J  a-  cos-  x  -f  b'-  sin-  x 

r       1  ,  ..  r  1  r  1  —  cos  X 

(5)     j ,  (e)     I ,         (f)     I 

J  a  +  6c(isx  «^  f<  +  '^ciisx-f-  csinx  J    1  -\-  sinx 


9.   SliDWthat  I  ii  fx.  ^  ((  +  bx  +  ex-)  maybe  treated  by  trigonometric  substitu- 
tion :  distinguish  between  b-  —  4  dc  ^  0. 


10.   Sliow  that   I  R(  X.  \'-^ )  is  made  rational  bv  w"  =  ~'^ —  •   Hence  infer 

J       \   '    \  rx  +  d/  '  rx  +  d 


"  i'lx  +  b\  .  ax  +  b 

\i I  is  made  rational  bv  y"  = 
ex  +  df                                  ■            ex  +  tZ 

that    I  /i(x.  A^(x  —  a)  (x  —  /3))  is  rationalized  bv  y-  =  "- This  accomplishes 

J  ■  X  —  a 

the  integration  of  7.'(x.  vo  +  bx  +  ex-)  when  the  roots  of  a  +  bx  +  rx-  =  0  are 

real,  that  is,  when  b-  —  4  ((c  >  0. 


24  INTRODUCTORY  REVIEW 

11     r.,  ,        r,-,  r      /ax  +  bX'"     lax-\-hY         1        ,  , 

11.  Show  that   I  U\  x.     1    .     ,  ■•  •    ,  where  the  exponents  m.  n, 

J      I  '  \cx  +  dj       \cx  +  d/  '        J 

•  •  •  are  rational,  is  rationalized  by  y^  = if  k  is  so  chosen  that  kin^  kn,  •  ■  •  are 

ex  +  d 
integers. 

12.  Show  that  f  {a  +  by)Pi/i  may  be  rationalized  if  p  or  g  or  p  +  ^  is  an  integer. 

By  setting  x"  =  y  show  that   /  x'"  (a  +  bx")  >'  may  be  reduced  to  the  above  type  and 

.    .  ,  ,       ,        m  +  1  ?H  +  1  .    .  , 

hence  is  mtei^rable  when or  p  or \-  p  is  integral. 

n  n 

13.  If  the  roots  of  a  +  hx  +  ex"  =  0  are  imaginary,    /  li  (x,  Va  +  hx  +  ex-)  may 
be  rationalized  by  ^  =  v  ({  +  bx  +  ex-  =F  -c  V c. 

14.  Integrate  the  following. 

•^Vx-l  -^l+vx  -^Vl  +  J^-Vl  +  x 

1 


(5)r-£L.,  (e)   f      J^ ,  (of 


V(l  — x'-)='  "^  (x  —  d)  Va  +  5x  +  ex- 


*^X(1  +  X2)2  ^  X  ^ 


V2  X-  +  X  ,  ^    r      a-^  ^1  —  a;^ 


Vl-x^ 


15.  In  view  of  Ex.  12  discuss  the  integrability  of  : 

,    s    r  ■  ,         .  r  .    C       ic™  r  let  X  =  ay-. 

(cr)    I  sin'"xcos"x,  let   sinx=Vy,  (/3)    I  — ^=    -^  

•^  *^   V'ax  —  X-     t  "^"     V  «x  -  X-  =  xy. 

16.  Apply  the  reduction  foruuihvs,  Table,  p.  60,  to  show  that  the  final  integral  for 

r      x'"  .         /"        1  r       X  r         ^ 

I  IS      I  — or      I  — r==z      or      I  ^z^^z; 

-J  Vl  -  X-  -^  Vl  -  X-  "^  Vl  -  X-  -^  X  Vl  -  X- 

according  as  ?rt  is  even  or  odd  and  positive  or  odd  and  negative. 

17.  Trove  sundrv  of  the  formulas  of  Peirce's  Table. 


18.  Show  that  if  A' (x,  Va-  —  x-)  contains  x  only  to  odd  powers,  the  substitu- 
tion z—Vd-  —  x-  will  rationalize  the  expression.  Use  Exs.  1  (f)  and  6  (e)  to 
comjiare  tlie  labor  of  this  algebraic  snbstitution  with  that  of  the  trigonometric  or 
hyperbolic. 

16.  Definite  integrals.  If  an  interval  from  .r  =  a  to  :'■  =:  Z*  be  divided 
into  n  successive  intervals  A./'i,  Xi\,  ■  ■  ■,  A./'„  and  the  value  /(^,)  of  a 
function /(.'■)  l)e  computed  from  some  point  ^,-  in  each  interval  A.j;,-  and 
be  multiplied  by  A./-,-,  then  t/ie  Ilinlt  of  the  snm 

lim  [/( t\)  A,/'i  +f(i,)  A./'o  +  •  •  •  +  /XU  A.e„]  =  f^  /(x)  dx.         (G2) 


FUNDAMENTAL  RULES 


25 


when  each  interval  becomes  infinitely  short  and  their  number  n  be- 
comes infinite,  is  known  as  the  definite  integral  oi  f{x)  from  a  to  h,  and 
is  designated  as  indicated.  If  y  =/(•?)  be  graphed,  the  sum  will  be 
represented  by  the  area  under 
a  broken  line,  and  it  is  clear 
that  the  limit  of  the  sum,  that 
is,  the  integral,  will  be  repre- 
sented by  'the  area  under  the 
curre  y  =f(.r)  and  between 
the  ordinates  :/:  =  a  and  ./•  =  b. 
Thus  the  definite  integral,  de- 
fined arithmetically  by  (G2), 
may  be  connected  with  a  geo- 
metric concept  which  can  serve  to  suggest  properties  of  the  integral 
much  as  the  interpretation  of  the  derivative  as  the  slope  of  the  tan- 
gent served  as  a  useful  geometric  representation  of  the  arithmetical 
definition  (2). 

For  instance,  if  a,  h,  c  are  successive  values  of  .r,  then 


£m'^-'-+IJm^-=!j(c'')d^ 


(63) 


is  the  equivalent  of  the  fact  that  the  area  from  a  to  c  is  equal  to  the 
sum  of  the  areas  from  a  to  L  and  //  to  r.  Again,  if  A.r  be  considered 
positive  when  .r  moves  from  a  to  />,  it  must  be  considered  negative 
when  X  moves  from  h  to  a  and  hence  from  (62) 


fj(x)dx  =  -Jj(,,^dx. 


(64) 


Finally,  if  J/  be  the  maximum  of  /{■>')  in  the  interval,  the  area  under 
tlie  curve  will  be  less  than  that  under  the  line  //  =  ^[  through  the 
highest  point  of  the  curve ;  and  if  in  he  the  minimum  of  /(.'),  the 
area  under  the  curve  is  greater  than  that  under  // =  ni.    Hence 


ra  (b  -  ^0  <  X  ■^'(•'■)  ^^^  '^  '^^(^'  ~  ")• 


(65) 


There  is,  then,  some  intermediate  value  ///  <  fj.<  ^f  such  that  the  inte- 
gral is  ec[ual  to  fi(h  —  a);  and  if  the  line  y  =  p.  cuts  the  curve  in  a 
point  whose  abscissa  is  ^  intermediate  between  a  and  b,  then 

jj(:r)  d:r  =  f.(b-  a)  =  (b  -  a)f(^).  (G5') 

This  is  the  fundamental  llieorem  of  the  Mean  for  definite  integrals. 


26  INTRODUCTORY  REVIEW 

The  definition  (62)  may  be  applied  directly  to  the  evaluation  of  the  definite  in- 
tegrals of  the  simplest  functions.  Consider  first  1/x  and  let  a,  b  be  positive  with  a 
less  than  b.  Let  the  interval  from  a  to  6  be  divided  into  n  intervals  Ax,  which  are 
in  geometrical  progression  in  the  ratio  r  so  that  Xi  =  a,  x-z  —  ar,  ■  •  •,  Xn+i  —  ar" 

and    Axi  =  a(?-  — 1),    Ax2  =  ar{r  —  I),  AX3  =  ar'-{r  —  1),  •  •  •,    Ax„  =  «r"-i(/' —  1) ; 

whence        b  —  a  —  Axi  +  Axo  +  •  •  •  +  XCn  —  «■  (/•"  —  1)     and     r"  =  b/a. 

Choose  the  points  |,-  in  the  intervals  Ax,-  as  the  initial  points  of  the  intervals.   Then 

Axi    ,   Ax.2  Ax„  _  a  (r  -  1)       ar{r-l)  (o-"-i  (/•-])  _ 

But  r=Vb/a     or     ji  =  log  (///«)  ^  log  r. 

Axi       Ax2                  Ax„  ,        , ,       .      b    r  —  I       .       b  h 

Hence 1 h  •  •  •  H =  n  (r  —  1)  =  log  - 


li         I2  f«  "    log/-  ''a    log(l  +  A) 

Now  if  71  becomes  infinite,  r  approaches  1,  and  li  approaches  0.  But  the  limit  of 
log  (1  +  Ji)/h  as  /i  zb  0  is  by  definition  the  derivative  of  log  (1  +  x)  when  x  =  0  and 
is  1.    Ilence 

I      —  =  hm      — '  -{ =  +  ---H '  \  =  hiir-  =  \oiib  —  loija. 

J  a     X       « =  =0  L  ?i         I2  l«  A  '(         '^ 

As  another  illustration  let  it  be  re(juired  to  evaluate  the  integral  of  cos-x  from 
0  to  I  IT.  Here  let  the  intervals  Ar,-  be  equal  and  their  number  odd.  Chuose  the  |"s 
as  the  initial  points  of  their  intervals.     The  sum  of  wliich  the  limit  is  desired  is 

(T  —  cos-  0  •  A.e  +  cos-  Ax  •  Ax  +  cos-  2  Ax  ■  Ax  +  •  •  • 

+  cos-  («  —  2)  Ax  ■  Ax.  +  cos-  [n  —  1)  Ax  •  Ax. 

But         nA.c  =  I  IT,  and  (»  —  1 )  Ax  =  ^  tt  —  Ax.   (//  —  2)  Ax  =  i  tt  —  2  Ax,  •  ■  •, 

and  cos  {\  tt  —  y)  =  sin  y     and     sin-  y  +  cos-  (/  =  1. 

Hence         cr  =  Ax  [cos-  0  +  cos-  Ax  +  cos-  2  A.C  +  •  •  •  +  .sin-  2  Ax  +  sin-  Ax] 

Hence  |    "  cos-xcZx  =  liin  [^5  nXc  +  ',  A.c]  =  lim  {\  rr  +  ]  Ax)  =  5  tt. 

Hidications  for  finding  the  integrals  nf  other  functions  are  given  in  the  exercises. 

It  should  be  noticed  that  the  variable  x  which  appears  in  the  exprcssidu  of  the 
definite  integral  really  has  nothing  to  do  with  the  value  of  the  integral  but  merely 
serves  as  a  symbol  useful  in  forming  the  sum  in  ((i2).  What  is  nf  importance  is 
the  function /and  the  limits  a.  b  of  the  interval  over  which  the  integral  is  taken. 

J  y (x)  <lr-=  f^  [fit)  at  =  J  ''f(y)  dy  =  J  /(*)  d*. 

The  variable  in  the  integrand  disappears  in  the  integration  and  leaves  the  value  of 
the  inteirral  as  a  functi(jn  of  the  limits  a  and  /;  alone. 


FUNDAMENTAL   KULES  27 

17.   If  the  lower  limit  of  the  integral  be  fixed,  the  value 


I 


of  the  integral  is  a  function  of  the  upper  limit  regarded  as  variable. 
To  find  the  derivative  ^'{f>),  form  the  quotient  (2), 


^(/>  + A/y)  -<!>( 


f{:r)<h--    {    f{,-)dx 


]^y  applying  (63)  and  (65'),  this  takes  the  simpler  form 


>h  +  Sb 


•A/.  "  A/y  -A//-^^^^-^' 

wliere  $  is  intermediate  l^etween  f/  and  /v  +  \h.  Let  A/^  =  0.  Then  i 
approaches  b  and/'(^)  ai)proaehes /'(T').    Hence 

^'C/0  =  |;J'/C^')^^-^-=/(^')-  (66) 

If  preferred,  the  varial)le  //  may  be  written  as  ./•,  and 

This  equation  will  establisli  the  relation  l)ctween  the  definite  integral 
and  the  indefinite  integral.  For  ];y  definition,  the  indefinite  integral 
7-'(,/-)  of /'(./•)  is  any  function  such  that  /''(•'')  equals /'(./•).  As  ^'(.'■)  =/{■>') 
it  follows  that  ^r 

I   /(..)./..  =  7-X.'0+C.  (67) 

Hence  except  for  an  additive  constant,  the  indefinite  integral  of  f  is 
the  definite  integral  of /' from  a  fixed  lower  limit  to  a  variable  u])per 
limit.  As  tlie  definite  integral  vanishes  when  the  upper  limit  coincides 
with  tlie  lower,  tlie  constant  C  is  —  /'X''')  and 


/' 


f{,^d,-^F(J.)-F(n).  (67') 


Hence,  iltr  dpfiiufe  IntegmJ  of  f(x)  from  a  to  h  is  flic  (Jlfferimce  hetireen 
tlie  valves  of  "ny  Indt'tinlfe  tnte'jrol  F{.r')  tohoi  for  tlie  uiiper  tnid  lover 
limits  of  the  di'fnite  biterjrdl ;  and  if  the  indefinite  integral  of  /  is 
known,  the  definite  integral  may  be  obtained  without  a})plying  the 
definition  (62)  to/ 


28  INTKODUCTOIiY  KEVIEW 

The  great  iinportancie  of  definite  integrals  to  geometry  and  physics 
lies  in  that  fact  that  many  quantities  connected  with  geometric  figures 
or  physical  bodies  mruj  he  e^iwessed  simply  for  small  portions  of  the 
figures  or  bodies  and  may  then  be  obtained  as  the  sum  of  those  quanti- 
ties taken  over  all  the  small  portions,  or  rather,  as  the  Vwiit  of  that  sum 
when  the  portions  become  smaller  and  smaller.  Thus  the  area  under  a 
curve  cannot  in  the  first  instance  be  evaluated  ;  l)ut  if  only  that  portion 
of  the  curve  which  lies  over  a  small  interval  A.r  be  considered  and  the 
rectangle  corresponding  to  the  ordinate /(^)  be  drawn,  it  is  clear  that 
the  area  of  the  rectangle  is  f(^)  A.r,  that  the  area  of  all  the  rectangles  is 
the  sum  2 /"(O  ^'''  taken  from  a  to  h,  that  when  the  intervals  Aa;  approach 
zero  the  limit  of  their  sum  is  the  area  under  the  curve  ;  and  hence  that 
area  may  be  Avritten  as  tl:e  definite  integral  of /(,?•)  from  <i  to  h* 

In  like  maimer  consider  the  mans  of  a  rod  of  variable  density  and  suppose  the 
rod  to  lie  along  the  x-axis  so  that  the  density  may  be  taken  as  a  function  of  x. 
In  any  small  length  Ax  of  the  rod  the  density  is  nearly  constant  and  the  mass  of 
that  part  is  approximately  equal  to  the  product  pAc  of  the  density  p{x)  at  the 
initial  point  of  that  part  times  the  length  Ax  of  the  part.  In  fact  it  is  clear  that 
the  mass  will  be  intermediate  between  the  products  /«Ax  and  3/Ax.  where  m  and 
M  are  the  miniimim  and  maxinmm  densities  in  the  interval  Ax.  In  other  words 
the  mass  of  the  st'ction  Ax  will  be  exactly  equal  to  p  (t)  Ax  where  f  is  .some  value  of 
X  in  the  interval  Ac.  The  mass  of  the  whole  rod  is  therefore  the  sum  2/3(^)Ax 
taken  from  one  end  of  the  rod  to  the  other,  and  if  the  intervals  be  allowed  to 
approach  zero,  the  mass  may  be  written  as  the  integral  of  p(x)  from  one  end  of 
the  rod  to  the  other,  t 

Another  problem  that  may  be  treated  by  these  methods  is  that  of  finding  the 
total  preHsurc  on  a  vertical  area  submerged  in  a  liciuid,  say,  in  water.  Let  w  be  the 
weight  of  a  column  of  water  of  cross  section  1  s(].  unit  and 
of  height  1  unit.  (If  the  unit  is  a  foot,  lo  =  62.5  lb.)  At  a 
point  h  units  below  the  surface  of  the  water  the  pressure  is 
idi  and  upon  a  small  area  near  that  depth  the  pressure  is 
approximately  idiA  if  A  be  the  area.  Tlie  pressure  on  the 
area  A  is  exactly  equal  to  iv^A  if  ^  is  some  depth  interme- 
diate between  that  of  the  top  and  that  of  the  bottom  of 
the  area.  Now  let  the  finite  area  be  ruled  into  strips  of  lielght  Ah.  Consider  the 
product  vjhb  {h)  A/i  where  h{h)  =f{h)  is  the  breadth  of  the  area  at  the  depth  h.   This 

*  The  ^'s  may  evidently  be  so  chosen  that  the  finite  sum  1f(^)\x  is  exactly  equal  to 
tlie  area  under  the  curve  ;  Ijut  still  it  is  iiccessarv  to  let  the  intervals  approach  zero  and 
thus  replace  the  sum  1)y  an  intejiral  because  the  values  of  ^  whicli  make  the  sum  equal 
to  the  area  are  unknown. 

t  This  and  similar  problems,  here  treated  hy  usins  the  Theorem  of  the  Mean  for 
integrals,  may  hu  treated  from  the  point  of  view  of  differentiation  as  in  §  7  or  from  that 
of  Didiamel's  or  Osgood's  Theorem  as  in  §§  "A,  .".t.  It  should  he  needless  to  state  that  in 
any  particular  problem  some  one  f)f  the  three  methods  is  likely  to  l)e  somewhat  preferable 
to  citiier  of  the  others.  The  reason  for  laying  such  emphasis  upon  the  Theorem  of  the 
Mean  here  and  in  the  exercises  lielow  is  that  the  theorem  is  in  itself  very  important  and 
needs  to  he  tlior(>UL:;hlv  mastered. 


FUNDAMENTAL  RULES  29 

is  approximately  tlie  pressure  on  tlie  strip  as  it  is  tlie  pressure  at  the  top  of  tlie  strip 
multiplied  by  the  approximate  area  of  the  strip.  Then  iv^b{^)Ah,  where  f  is  some 
value  between  h  and  h  +  Ah,  is  the  actual  pressure  on  the  strip.  (It  is  sufficient  to 
write  the  pressure  as  approximately  whb{h)Ah  and  not  trouble  with  the  ^.)  The 
total  pressure  is  then  2w^&(?)  Ah  or  better  the  limit  of  that  sum.    Then 


P  =  lim  V?<.'?6(^)d/i  =  r  i€hb{h)dh, 


wliere  a  is  the  depth  of  the  toi)  of  the  area  and  b  that  of  the  bottom.  To  evaluate 
the  pressure  it  is  merely  necessary  to  find  the  breadth  5  as  a  function  of  h  and 
integrate. 

EXERCISES 

kf{x)dx  =  k  I   f{x)dx. 

a  *j  n 

Xh  p  b  /%  b 

(u  i:  v)  dx  —  I    udx  ±  j    vdx, 

\p{x)dx  <  /   f{x)dx  <  I    (p{x)dx. 

a  ^a  on 

4.  Suppose  that  the  minimum  and  maximum  of  the  quotient  Q{x)  =f{x)/(p{x) 

of  two  functions  in  the  interval  from  «  to  6  arc  m  and  M,  and  let  0  (x)  be  positive 

so  that 

m  <  Q  (.c)  =  -^  <  M    anil     m4>  (x)  <f{x)  <  Mc}>  (x) 
<p(x) 

are  true  relations.    Show  by  Exs.  3  and  1  that 

f''f{x)dx  f''f{x)dx 

m  <  — <  M    and     ^ =  fj.=  0^)  =  ■^-^'- , 

/    cf>{x)dx  /   cp{x)dx  ^^^' 

where  ^  is  some  value  of  x  between  a  and  b. 

5.  If  m  and  3/  are  the  minimum  and  maximum  of  f{x)  between  a  and  b  and  if 
<p  (x)  is  always  positive  in  the  interval,  show  that 

(p  (x)  dx  <  I    /(x)  4>  (x)  dx  <  M  I    <p  (x)  dx 

and  (  /(x)  <t>  (x)  dx  =  fj.  f  0  (x)  dx  =f{^)  f  <P  ('C)  dx. 

Note  that  the  integrals  of  [M  —  f  {x)]  (p  (x)  and  [/(x)  —  ?«]  0  (x)  are  positive  and 
apply  Ex.  2. 

6.  Evaluate  the  following  by  the  direct  application  of  (62)  : 

J"^             b"  —  ffl^  1^  ^ 

xdx  = ,  (/3)     I    e^dx  =  t*  —  e". 

a  2  «/ a 

Take  equal  intervals  and  use  the  rules  for  arithmetic  and  geometric  progressions. 

7.  Evaluate  {a)    f  x"'dx  = (?>"'  +  i  -  «"'+!),       (^)    f  r'dx  = (c*  -  c°). 

Jn  in  -\-  1  J„  logc 

In  the  first  the  intervals  should  be  taken  in  geometric  progression  with  r"  =  l>/a. 


30  INTRODUCTORY   REVIEW 

siii^xdx  =  \  TT,    (/3)    I     coii"xdx  =  0,  if  n  is  odd. 

0  "Jo 

9.  With  the  aid  of  tlie  trigonometric  formulas 

cosx  +  cos  2  X  +  •  •  •  +  cos  (((  —  1)  X  =  I  [sin  iix  cot  J  x  —  1  —  eos?w;], 
sinx  +  sin  2x  +  •  •  •  +  sin  (?t  —  1)  x  =  h  [(1  —  cosnx)  cot  J  x  —  sin?u;], 

J-»  ?>  /^  h 

cosxdx  =  sin6  —  sin  a,  (;3)    /    sinxrZx  =  cosa  —  cos/;. 

a  ^  it 

10.  A  function  is  said  to  be  men  if  /(—  x)  =/(x)  and  odd  if  /(—  x)  =  —f{x) 
Show    («)    r     7(x)(7x  =  2   r7(x)cZx, /even,         (/3)    f     7(x)(Zx  =  0, /odd. 

%J  ~~  a  */  0  J  —  a 

11.  Show  that  if  an  integral  is  regarded  as  a  function  of  the  lower  limit,  the 
upper  limit  being  fixed,  then 

*»  =  T-    Cf{x)dx  =  -f{a),       if    <f>(a)=  r/(x)rZx. 
da  J  a  J, I. 

12.  Use  the  relation  between  definite  and  indefinite  integrals  to  compare 
/(x)  dx  =  {b  -  a)f{^)     and     F{h)  -  F{a)  =  {h  -  a)  F'(f), 


•/a 


the  Theorem  of  the  Mean  for  derivatives  and  for  dcfiinte  integrals. 

13.  From  consideration  of  Exs.  12  and  4  establish  Cauchy'a  Formnlu 

which  states  that  the  quotient  of  the  increments  AF  and  A<[»  of  two  functions,  in 
any  interval  in  which  the  derivative  4>'(x)  does  not  vanish,  is  equal  to  the  quotient 
of  the  derivatives  of  the  functions  for  some  interior  point  of  the  interval.  What 
would  the  application  of  the  Theorem  of  the  Mean  for  derivatives  to  numerator 
and  denominator  of  the  left-hand  fraction  give,  and  wherein  does  it  differ  from 
Cauchy's  Fornuda  ';* 

14.  Discuss  tlu!  volume  of  revolution  of  y  =/(x)  as  the  linnt  of  tlitssum  of  thin 
cylinders  and  compare  the  results  with  thf)se  found  in  Ex.  0,  \).  ](.). 

15.  Show  that  the  mass  of  a  rod  running  from  a  to  /;  along  the  x-axis  is 
I  k(l)^  —  a")  if  the  density  varies  as  the  distances  from  the  origin  [k  is  a  factor  of 
proportionality). 

16.  Sh<>w  (a)  that  the  mass  in  a  rod  running  from  rt  to  h  is  i  lie  same  as  the  ai'ca 
under  the  curve  y  =  p  (x)  between  the  ordinates  x  =  a.  and  x  =  h.  and  explain  why 
this  should  be  seen  intuitively  to  be  so.  Show  (/3)  that  if  the  density  in  a  plane  slab 
bounded  by  the  x-axis,  the  curve  y  =/(x),  and  the  ordinates  x  =  '/,  and  x  =  h  is  a 

function  p  (x)  of  x  alone,  the  mass  of  the  slab  is  I  yp  (x)  dx  ;  also  (7)  that  the  mass 
of  the  corresponding  volume  of  revolution  is    /    iry'-p(x)dx. 

17.  An  isosceles  triangle  has  the  altitude  (t,  and  the  base  '2h.  Find  (cr)  the  mass 
on  the  assumption  that  the  density  varies  as  the  distance  from  the  vertex  (meas- 
ured along  the  altitiide).  Find  (/3)  the  mass  of  the  cone  of  revolution  formed  by 
revolving  the  triam;le  about  its  altitude  if  the  law  of  densitv  is  the  same. 


FUXD A:\rENTAL   RULES  31 

18.  In  a  plane,  the  moment  of  inertia  I  of  a  particle  of  mass  m  with  respect  to  a 
point  is  defined  as  the  product  mr-  of  the  mass  by  the  square  of  its  distance  from  tlie 
point.    Extend  this  definition  from  particles  to  bodies. 

{a)  Show  tliat  tlie  moments  of  inertia  of  a  rod  running  from  a  to  b  and  of  a 
circular  slab  of  radius  a  are  respectively 

I  =   I    x'-p  {x)  dx     and     1=1    2  Trr'p  (?•)  clr,         p  the  density, 

if  the  point  of  reference  for  the  rod  is  the  origin  and  for  the  slab  is  the  center. 

(j3)  Show  that  for  a  rod  of  length  21  and  of  uniform  density,  1=  \M1^  with 
respect  to  tlie  center  and  I  =  \  3/f-  with  respect  to  tlie  end,  J/  being  the  total  mass 
of  the  rod. 

(7)  For  a  uniform  circular  slab  with  respect  to  the  center  I  =  ^  J/a'-. 

(5)  For  a  uniform  rod  of  length  21  witli  respect  to  a  point  at  a  distance  d  from 
its  center  is  I  =  ^I {\  I"  +  d-).  Take  the  rod  along  the  axis  and  let  the  point  be 
(tr,  /3)  with  d~  =  a-  +  /3-. 

19.  A  rectangular  gate  holds  in  check  the  water  in  a  reservoir.  If  the  gate  is 
submerged  over  a  vertical  distance  II  and  has  a  breadth  B  and  the  top  of  the 
gate  is  ft  units  below  the  surface  of  the  water,  find  the  pressure  on  the  gate.  At 
what  depth  in  the  water  is  the  point  where  the  pressure  is  the  mean  pressure 
over  the  gate  '? 

20.  A  dam  is  in  the  form  of  an  isosceles  trapezoid  100  ft.  along  the  top  (which 
is  at  the  water  level)  and  00  ft.  along  the  botttmi  and  30  ft.  high.  Find  the  pres- 
sure in  tons. 

21.  Find  the  pressure  on  a  circular  gate  in  a  water  main  if  the  radius  of  the 
circle  is  r  and  the  depth  of  the  center  of  the  circle  below  the  water  level  is  d^r. 

22.  In  space,  moments  of  inertia  are  defined  relative  to  an  axis  and  in  the  for- 
mula I  =  mr-,  for  a  single  particle,  r  is  the  perpendicular  distance  fr(.)ni  the 
partit'le  to  the  axis. 

(a)  Show  that  if  the  density  in  a  solid  of  revolution  generated  by  ij  =/(,/•)  varies 
only  with  the  distance  along  the  axis,  the  moment  of  inertia  about  the  axis  of 

revolution  is  7  =    I     I  Trt/^p{x)dx.    Apply  Lx.  18  after  dividing  tlie  solid  into  disks. 

(/3)  Find  the  moment  of  inertia  of  a  sphere  about  a  diameter  in  case  the  density 
is  constant ;  I  —  i  Ma-  =  ^-^  Trprr"'. 

(7)  Apply  the  I'esult  to  find  the  moment  of  inertia  of  a  spherical  shell  with 
external  and  internal  radii  a  and  h  :  I  =  =  M{a''  —  lr)/(a"  —  //").  Let  h  =  a  and 
thus  find  1=  '^ISIC'  as  the  moment  of  inertia  of  a  spherical  surface  (shell  of  negli- 
gible thickness). 

(5)  For  a  cone  of  revolution  I  —  j^,j  Ma"  where  a  is  the  radius  of  the  base. 

23.  If  the  force  of  attraction  exerted  l)y  amass  m  upon  a  point  is  lnnf{r)  where 
r  is  the  distance  f  nim  the  mass  to  the  point,  show  that  the  attraction  exerted  at 
the  origin  l)y  a  rod  of  density  p(x)  running  from  a  to  b  along  tlie  x-axis  is 

A  =  f  lf{x)  p  (x)  dx,     and  that     A  =  kJI/ab,         M  =  p{b-  a), 

is  the   attraction  of   a  uniform  rod  if  the  law  is   the  Law  of  Nature,   that  is, 


32  mTKO])UCTOKY   KEVIEW 

24.  Suppose  that  the  density  p  in  the  slab  of  Ex.  16  were  a  function  p  (x,  y)  of 
both  X  and  y.  Sliow  that  the  mass  of  a  small  slice  over  the  interval  Ax,-  would  be 
of  the  form 

Ac  I  p{x,  y)dy  = 'i>{^)Ax   and  that     I    ^  (,;;)  Ax  =  I         |  p{x,y)dy\clx 

would  be  the  expression  for  the  total  mass  and  \v((uld  recpiire  an  integration  with 
respect  to  y  in  which  x  was  held  constant,  a  suljstitution  of  the  limits  f{x)  and  0 
for  y,  ajid  then  an  integration  with  respect  to  x  from  a  to  b. 

25.  Apply  the  considerations  of  Ex.  24  to  finding  moments  of  inertia  of 
[a)   a  uniform  triangle  y  =  nix,  y  —  0,  x  =  a  with  respect  to  the  origin, 
(/3)  a  uniform  rectangle  with  respect  to  the  center, 

(7)  a  uniform  ellipse  with  respect  to  the  center. 

26.  Comimre  Exs.  2-4  and  10  to  treat  the  volume  under  the  surface  2  =  p  (x,  y) 
and  over  the  area  liounded  by  y  =f(x),  ?/  =  0.  x  =  a,  x  =  h.    Find  the  volume 

(or)  under  z  =  xy  and  over  y-  =  4px.  y  =  0.  x  =  0.  x  =  h, 

{^)  luider  z  =  X-  +  y-  and  over  x-  +  y'^  =  n-.  y  =  0.  x  —  0.  x  =  Q. 

(7)  under  '—  +  —  +  -  =  1  and  over  --  +  —  =  1,  ?/  =  0,  x  =  0,  x  =  a. 
a-       Ir      c-  a-      b- 

27.  Discuss  sectorial  area  4  I  r-d4)  in  polar  coordinates  as  the  limit  of  the  sum 
of  small  sectors  running  out  from  the  pole. 

28.  Show  that  the  moment  of  inertia  of  a  uniform  circular  sector  of  angle  a 
and  radius  a  is  \  pan*.    Hence  infer  I  =  \  p  i      r^d(p  in  polar  coordinates. 

Jaf, 

29.  Find  the  moment  of  inertia  of  a  uniform  (a)  lemniscate  ;•"  =  a-  cos-  2  (p 
and  (^)  cardioid  r  =  a  (1  —  cos0)  with  respect  to  the  pole.  Also  of  (7)  the  circle 
r  =  2  a  cos  <p  and  (5)  the  rose  r  =  a  sin  2  0  and  (e)  the  rose  r  =  a  sin  3  0. 


CHAPTER   II 

REVIEW  OF  FUNDAMENTAL  THEORY* 

18.  Numbers  and  limits.  The  concept  and  theory  of  real  number, 
integral,  rational,  and  irrational,  will  not  be  set  forth  in  detail  here. 
Some  matters,  however,  which  are  necessary  to  the  proper  understand- 
ing of  rigorous  methods  in  analysis  must  be  mentioned ;  and  numerous 
points  of  view  which  are  adopted  in  the  study  of  irrational  nund)er 
will  be  suggested  in  the  text  or  exercises. 

It  is  taken  for  granted  that  by  his  earlier  work  the  reader  lias  become  familiar 
with  the  use  of  real  numbers.  In  particular  it  is  assumed  that  he  is  accustomed 
to  represent  numbers  as  a  scale,  that  is,  by  points  on  a  straight  line,  and  that  he 
knows  that  when  a  line  is  given  and  an  origin  chosen  upon  it  and  a  unit  of  measure 
and  a  positive  direction  have  been  chosen,  then  to  each  point  of  the  line  corre- 
sponds one  and  only  one  real  number,  and  conversely.  (.)\ving  to  this  correspond- 
ence, that  is,  owing  to  the  conception  of  a  scale,  it  is  possible  to  interchange 
statements  about  numbers  with  statements  about  points  and  hence  to  obtain  a 
more  vivid  and  graphic  or  a  more  abstract  and  arithmetic  phraseology  as  may  be 
desired.  Thus  instead  of  saying  that  the  numbers  Xi,  X2,  •  •  •  are  increasing  algebra- 
ically, one  may  say  that  the  points  (whose  coordinates  are)  Xi,  X2,  •  •  •  are  moving 
in  the  positive  direction  or  to  the  right ;  with  a  similar  correlation  of  a  decreasing 
suite  of  numbers  with  points  moving  in  the  negative  directiun  or  to  the  left.  It 
should  be  remembered,  however,  that  whether  a  statement  is  couched  in  geometric 
or  algebraic  terms,  it  is  always  a  statement  concerning  numbers  when  one  has  in 
mind  the  point  of  view  of  pure  analysis,  t 

It  may  be  recalled  that  arithmetic  begins  with  the  integers,  including  0,  and 
with  addition  and  multiplication.  That  second,  the  rational  numbers  of  the 
foi"m  p/q  are  introduced  with  the  operation  of  division  and  the  negative  rational 
numbers  with  the  operation  of  subtraction.  Finally,  the  irrational  numbers  are 
introduced  by  various  processes.  Thus  V2  occurs  in  geometry  through  the 
neces.sity  of  expressing  the  length  of  the  diagonal  of  a  square,  and  V3  for  the 
diagonal  of  a  cube.  Again,  tt  is  needed  for  the  ratio  of  circumference  to  diameter 
in  a  circle.  In  algebra  any  equation  of  odd  degree  has  at  least  one  real  root  and 
hence  may  be  regarded  as  defining  a  number.  But  there  is  an  essential  dilference 
between  rational  and  irrational  numbers  in  that  any  rational  number  is  of  the 

*  Tlie  object  of  tliis  chapter  is  to  set  forth  systematically,  witli  attention  to  precision 
of  statemeiU  and  accuracy  of  profif,  those  fundamental  definitions  and  tlieorems  which 
lie  at  the  basis  of  calculus  and  whicli  have  been  given  in  tlie  previous  chapter  from  an 
iutuitive  rather  tliaii  a  critical  point  of  view. 

t  Some  ilhistrative  graphs  will  be  given;  tlie  student  should  make  many  others. 


34  INTllODUCTOKY   REVIEW 

form  ±  p/q  with  q  j^  0  and  can  therefore  be  written  down  explicitly  ;  whereas 
tlie  irrational  numbers  arise  by  a  variety  of  processes  and,  altlioutrh  they  may  be 
represented  to  any  desired  accuracy  by  a  decimal,  they  cannot  all  be  written 
down  explicitly.  It  is  therefore  necessary  to  have  some  definite  axioms  regulating; 
the  essential  properties  of  irrational  luimbers.  The  particular  axiom  upon  which 
stress  will  here  be  laid  is  the  axl(jm  of  contiiniity,  the  use  of  which  is  essential 
to  the  proof  of  elementary  theorems  on  limits. 

19.  Axiom  of  Coxtixi'Ity.  Jfa//  f/tej/oinfs  of(i  Ihie  are  ilirl/Icd  into 
tiro  classes  siiclt  tliaf  crenj  ji<}lnf  of  flw  fi rst  class  jtreeedes  evert/  paint  of 
the  second  class,  there  must  he  a  jiolnt  C  siicJi  that  any  point  preeedimi 
C  is  in  til e  first  class  and  a ni/  point  succeed inij  C  is  in  the  second  class. 
Tliis  principle  may  l^e  stated  in  terms  of  mimbers,  as  :  If  all  real  num- 
bers he  assorted  into  tico  classes  such  that  crenj  numhcr  (f  the  first  class 
is  alfjehraicalhj  less  than  every  ninnhcr  (fi  the  second  class,  tJwre  ruust  he 
a  numhcr  X  such  that  any  numhcr  less  thmi  X  is  in  tin'  first  class  and 
any  numhcr  yreater  tJian  X  is  in  the  second.  The  numljer  J\'  (Or  point  C) 
is  called  the  frontier  numl;)er  (or  point),  or  sim^ily  the  frontier  of  the 
two  classes,  and  in  particnlar  it  is  the  vpper  frontier  for  the  first  class 
and  the  lou-er  frontier  for  the  second. 

To  consider  a  particular  case,  let  all  the  negative  numbers  and  zero  constitute 
the  first  class  and  all  the  positive  numbers  the  second,  or  let  the  negative  luunbers 
alone  be  the  first  class  and  the  positive  numbers  with  zero  the  .second.  In  either 
case  it  is  clear  that  the  classes  satisfy  the  conditions  of  the  axiom  and  that  zero  is 
the  frontier  number  such  that  any  lesser  number  is  in  the  first  class  and  any 
greater  in  the  second.  If.  however,  one  were  to  consider  the  .system  f)f  all  positive 
and  negative  numbers  but  without  zero,  it  is  clear  that  there  wmdd  be  no  number 
X  which  would  satisfy  the  conditions  demanded  by  the  axiom  when  the  two 
classes  were  the  negatixc  and  jiositive  numbers  ;  for  no  matter  how  small  a  jiosi- 
tive  numlier  were  taken  as  X.  tliere  would  be  smaller  numbers  which  would  also 
be  positive  and  would  not  belong  to  the  first  class  ;  and  sinularly  in  case  it  were 
attempted  to  find  a  negative  X.  Thus  the  axiom  Insures  tlie  presence  of  zero  in 
the  system,  and  in  like  maniuu'  insures  the  presence  of  every  other  nundier  —  a 
matter  which  is  of  importance  because  there  is  no  way  of  writing  all  (irrational) 
numbers  in  explicit  form. 

Further  to  appreciate  the  continuity  of  the  niunber  scale,  consider  the  four 
significations  attributable  to  the  phrase  "  </(C  interval  from  u  to  h."    They  are 

«  =  X  s  h,         a  <  X  =  h.         a  =  X  <h.         (I  <  X  <  h. 

That  is  to  say,  both  end  points  or  either  or  neither  may  belong  to  the  interval.  In 
the  case  a  is  absent,  the  interval  has  no  first  point  ;  and  if  h  is  absent,  there  is  no 
last  point.  Tims  if  zero  is  not  counted  as  a  positive  lunnber.  there  is  no  least 
positive  nundier  ;  for  if  any  least  luunber  were  named,  half  of  it  would  surely  l)e 
less,  and  hence  the  absurdity.  The  axiom  of  continuity  shows  that  if  all  numbers 
lie  divided  into  two  classes  as  reijuired.  there  nuist  be  either  a  greatest  in  the  tirst 
class  or  a  least  in  the  second  —  the  frontier  —  but  not  both  unless  the  frontier  is 
counted  twice,  once  in  each  class. 


FUNDAMENTAL  TllEOEY  35 

20.  a)EFixiTiox  OF  A  Limit.    Ifx  is  a  va viable  vJilrh  takes  on  succes- 
sice  (•((lues  x^,  ,/■„  ■  •  •,  ,r,,  .)•/,  ■  ■  ■,  the  var'uihle  x  is  said  to  approach  the  con- 
stant I  as  a  11  III  It  If  tlie  ntniierlcal  difference  heticcen  x  and  I  ultliiiateJ ij 
heconies,  and  for  all  sacecedliuj  values  of  x  remains, 
less   tlian   any  preasshined  ninnhev  no  inattev   how       k    \  ly.   L    ''"'"'  '  ™ 
small.    The  numerical  difference  between  x  and  I 
is  denoted  by  |.''  — /|  or  \l  —  x\  and  is  called  the  absolute   value  of  the 
difference.    Tlie  fact  of  the  approach  to  a  limit  may  be  stated  as 

|.''  —  /|  <  e      for  all  .r's  subsequent  to  some  x 
or  X  =  I  -{-  7],      [77I  <  £      for  all  x's  subsequent  to  some  x, 

where  e  is  a  ])0sitive  number  which  niay  be  assigned  at  pleasure  and 
must  be  assigned  before  the  attemi)t  be  made  to  find  an  x  such  that 
for  all  subsequent  .r's  the  relation  |./'  —  /]  <  e  holds. 

So  long  as  the  conditions  required  in  tlie  definition  of  a  Hniit  are  satisfied  there 
is  no  need  of  bothering  about  liow  the  variable  approaches  its  linnt,  whether  from 
one  side  or  alternately  from  one  side  and  the  other,  whether  discontinuously  as  in 
the  case  of  the  area  of  the  polygons  used  for  computing  the  area  of  a  circle  or 
continuously  as  in  the  case  of  a  train  brouglit  to  rest  by  its  brakes.  To  speak 
geometrically,  a  point  x  which  changes  its  ixisitinu  upon  a  line  approaches  the 
point  I  as  a  limit  if  the  point  x  ultimately  comes  into  and  remains  in  an  assigned 
interval,  no  matter  how  small,  surrounding  I. 

A  variable  is  said  to  become  Infinite  if  the  numerical  value  of  the 
variable  idtimately  l)econies  and  remains  greater  than  any  preassigned 
number  A',  no  matter  how  large.*  The  notation  is  .r  =  cc,  but  had  best 
be  read  "  .r  becomes  infinite,"  not  "'  x  equals  inhnity." 

TiiEOKEM  1.  If  a  variable  is  always  increasing,  it  either  becomes 
infinite  or  approaches  a  limit. 

That  the  vaiiable  may  increase  indefinitely  is  apparent.  But  if  it  does  not 
become  infinite,  there  must  be  numbers  K  which  are  greater  than  any  value  of 
the  variable.  Then  any  number  nuist  satisfy  one  of  two  conditions:  either  there 
are  values  of  the  variable  which  are  greater  than  it  or  there  are  no  values  of  the 
variable  greater  than  it.  Mort'over  all  numbers  that  satisfy  the  first  condition  are 
less  than  any  number  which  satisfies  the  second.  All  minibers  are  therefore 
divided  into  two  classes  fulfilling  the  reciuirements  of  the  axiom  of  contiiuiity,  and 
there  must  be  a  number  .Y  such  that  there  are  values  of  the  variable  greater  than 
any  number  N  —  e  which  is  less  than  .Y.  Hence  if  e  be  assigned,  there  is  a  value  of 
the  variable  which  lies  in  the  interval  A^  —  e  <  x  ^  i\^,  and  as  the  variable  is  always 
increasing,  all  subsequent  values  nuist  lie  in  thi.s  interval.  Therefore  the  variable 
approaches  N  as  a  linut. 

*  This  (letiuition  means  what  it  says,  and  no  more.  Later,  additional  or  different 
meanings  may  l)e  assigned  to  infinity,  but  not  now.  Loose  and  extraneous  concepts  in 
this  connection  are  ahnost  certain  to  introduce  errors  and  confusion. 


36  INTRODUCTORY   REVIEW 

EXERCISES 

1.  If  Xi,  X2,  •  •  •,  x,„  •  •  •,  x„4.;„  ■  ■  •  is  a  suite  approacliing  a  limit,  apply  the  defi- 
nition of  a  limit  to  show  that  when  e  is  given  it  must  be  possible  to  find  a  value  of 
n  so  great  that  \Xn  +  p  —  Xn\<e  for  all  values  of  p. 

2.  If  Xi,  X2,  •  •  •  is  a  suite  approaching  a  limit  and  if  2/1,  2/2,  ••  •  is  any  suite  such 
that  \y„  —  x„ I  approaches  zero  when  n  becomes  infinite,  show  that  the  y''s  approach 
a  limit  which  is  identical  with  the  limit  of  the  x's. 

3.  As  the  definition  of  a  limit  is  phrased  in  terms  of  inequalities  and  absolute 
values,  note  the  following  rules  of  operation  : 

/N-r,.  ^  ,  ,,  c       h  ^     a      a 

(a)  It     a  >  0     and     c  >  b,    then     -  >  -     and     -  <  -  , 

a      a  c      b 

(^)  \a  +  b  +  c+  ...\s\a\  +  \b\  +  \c\+  •••,        {7)  |ate  •  •  .|  =  |a|.|6|.|c|.  •  ., 

where  the  equality  sign  in  (/3)  holds  only  if  the  luimbers  a,  b,  c,  ■  ■  ■  have  the  same 
sign.  By  these  relations  and  the  definition  of  a  limit  prove  the  fundamental 
theorems  : 

If  Unix  =  A'  and   lim  y  =  1',   then   lim  (x  ±  y)  =  X  ±  Y  and   lim  xy  =  XY. 

4.  Prove  Theorem  1  when  restated  in  the  slightly  changed  form  :  If  a  variable 
X  never  decreases  and  never  exceeds  7i,  then  x  approaches  a  Unfit  -A''  and  N  ^  K. 
Illustrate  fully.  State  and  prove  the  corresponding  theorem  for  the  case  of  a 
variable  never  increasing. 

5.  If  Xi,  x„,  ■  •  •  and  yi,  2/2,  ••  •  are  two  suites  of  which  the  first  never  decreases 
and  the  second  never  increases,  all  the  ?/\s  being  greater  than  any  of  the  x's,  and  if 
when  e  is  assigned  an  n  can  be  found  such  that  ?/„  —  x„  <  e,  show  that  the  limits 
of  the  suites  are  identical. 

6.  If  Xi,  x.„  •  •  •  and  ?/i,  2/2,  ••  •  are  two  suites  which  never  decrease,  show  by  Ex.  4 
(not  by  Ex.  3)  that  tlie  suites  xi  +  y-i,  X2  +  2/2,  •  •  •  and  Xj?/i,  X22/2,  •  •  •  ajiproach 
limits.  Note  that  two  iidiiute  decimals  are  precisely  two  suites  which  never  de- 
crease as  more  and  more  figures  are  taken.  They  do  not  always  increase,for  some 
of  the  figures  may  be  0. 

7.  If  the  word  "  all  "  in  the  hypothesis  of  the  axiom  of  continuity  be  assumed  to 
refer  only  to  rational  mnnbers  so  that  the  statement  becomes  :  If  all  rational 
numbers  be  divided  into  two  classes-  •  •,  there  sliall  be  a  number  N  (not  neces- 
sarily rational)  such  that  •  •  •  ;  tlien  the  conclusion  may  be  taken  as  defining  a 
lumiber  as  the  frontier  of  a  seciuence  of  rational  luimbers.  Show  that  if  two  num- 
bers X,  Y  be  defined  by  two  such  sequences,  and  if  tlui  sum  of  the  numbers  be 
defined  as  the  number  defined  by  the  se(iuence  of  the  sums  of  corresponding  terms 
as  in  Ex.  6,  and  if  the  product  of  the  numbers  be  defined  as  the  miniber  defined  l)y 
the  secjuence  of  the  products  as  in  Ex.  0,  then  th(!  fundamental  rules 

X+  Y=:Y+  X,  XY  =  YX,  {X  +  Y)Z  =  XZ  +  YZ 

of  arithmetic  hold  for  the  numbers  X,  Y,  Z  defined  by  sequences.  In  this  way  a 
conqik'te  theory  of  irrationals  may  be  built  up  from  the  properties  of  rationals 
combined  with  the  ])rinciple  of  (■(nitiiniity,  namely,  1°  by  defining  irrationals  as 
frontiei's  of  sefiucnces  of  I'ationals,  '2°  by  defining  tlie  operations  of  addition,  nuilti- 
l)licati()ii,  •  •  •  as  operations  upon  the  rational  numbers  in  the  scijuences,  3°  by 
showiii'^  that  the  fundamental  rules  of  arithmetic  still  hold  for  the  irrationals. 


fuxi)a:\iextal  theory  37 

8.  Apply  the  principle  of  continuity  to  show  that  there  is  a  positive  number  x 
such  that  z-  =  2.  To  do  this  it  should  be  shown  that  the  rationals  are  divisible 
into  two  classes,  those  whose  square  is  less  than  2  and  those  whose  square  is  not 
less  than  2  ;  and  that  these  classes  satisfy  the  requirements  of  the  axiom  of  conti- 
nuity. In  like  manner  if  a  is  any  positive  number  and  n  is  any  positive  integer, 
show  that  there  is  an  x  such  that  x"  =  a. 

21.  Theorems  on  limits  and  on  sets  of  points.  The  theorem  on 
limits  which  is  of  fundamental  algebraic  importance  is 

Theoke.m  2.  If  R  (x,  I/,  z,  •  •  •)  be  any  rational  function  of  the  variables 
X,  I/,  z,  •••,  and  if  these  variables  are  approaching  limits  A',  Y,  Z,  ■■■, 
then  the  value  of  Jl  approaches  a  limit  and  the  limit  is  7^  (.Y,  }',  Z,  ■  ■  •), 
provided  there  is  no  division  by  zero. 

As  any  rational  expression  is  made  up  from  its  elements  by  combinations  of 
addition,  subtraction,  multiplication,  and  division,  it  is  sufficient  to  prove  the 
theorem  for  these  four  operations.  All  except  the  last  have  been  indicated  in  the 
above  Ex.  3.  As  multiplication  has  been  cared  for,  division  need  be  considered 
only  in  the  simple  case  of  a  reciprocal  1/x.  It  must  be  proved  that  if  lim  x  =  A', 
then  lim  (1/x)  =  1/X.    N(jw 

t-Xl 


1        1 
x~X 


!A' 


by  Ex.  3  (7)  above. 


This  quantity  must  be  shown  to  be  less  than  any  assigned  e.    As  the  quantity  is 

complicated  it  will  be  replaced  by  a  simpler  one  which  is  greater,  owing  to  an 

increase  in  the  denominator.    Since  x  :^  X,  x  —  A'  may  be  made  numerically  as 

small  as  desired,  say  less  than  e',  for  all  x's  subsequent  to  some  particular  x.    Hence 

if  e'  be  taken  at  least  as  small  as  l\X\,  it  appears  that  |x]  must  be  greater  than 

i|A'|.    Then 

|x-  A"  I      |x-  A'j  e'  1      T^       o  /    X     1 

< = ,  bv  Ex.  3  (a)  above. 


|x||X|        l\X\-^        i|A|^ 

and  if  t'  be  restricted  to  being  less  than  ^|  A'|-e,  the  difference  is  less  than  e  and 
the  theorem  that  lim  (1/x)  =  1/A'  is  proved,  and  also  Theorem  2.  The  necessity 
for  the  restriction  A'  ^i  0  and  the  corresponding  restriction  in  the  statement  of 
the  theorem  is  obvious. 

Theorem  3.  If  when  e  is  given,  no  matter  how  small,  it  is  possil)le 
to  find  a  value  of  ?i  so  great  that  the  difference  |.''„  +  p  —  .'■„!  between  »■„ 
and  every  subsequent  term  .>'„4-/,  in  the  suite  :i\,  x^,  ■■■,  a'„,  ■••  is  less 
than  e,  the  suite  approaches  a  limit,  and  conversely. 

The  converse  part  has  already  l)ecn  given  as  Ex.  1  above.  The  theorem  itself  is 
a  consequence  of  the  axiom  of  continuity.  First  note  that  as  \x„j.p  —  .rn  |  <  e  for 
all  x's  subsequent  to  x„,  the  x"s  cannot  become  infinite.  Suppose  P  that  there 
is  some  number  I  such  that  no  matter  how  remote  x„  is  in  the  suite,  there  are 
always  subsequent  values  of  x  which  are  greater  than  /  and  others  which  are  less 
than  /.  As  all  the  x"s  after  x„  lie  in  the  interval  2e  and  as  /  is  less  than  some  x"s 
and  greater  than  others,  I  must  lie  in  that  interval.    Hence  \l  —  Xn^p\  <  2  e  for  all 


38  IXTK(3DUCT011Y  REVIEW 

x's  subsequent  to  x„.  But  now  2  e  can  be  made  as  small  as  desired  because  e  can  be 
taken  as  small  as  desired.  Hence  tlie  definition  of  a  limit  applies  and  the  x's 
approach  I  as  a  limit. 

Suppose  2°  that  there  is  ]io  such  number  I.  Then  every  number  k  is  such  that 
either  it  is  possible  to  go  so  far  in  the  suite  tliat  all  subsequent  numbers  x  are 
as  great  as  k  or  it  is  possible  to  go  so  far  that  all  subsequent  z's  are  less  than  k. 
Hence  all  numbers  k  are  divided  into  two  classes  which  satisfy  the  requirements  of 
the  axiom  of  continuity,  and  there  must  be  a  number  -N"  such  that  the  x'a  ultimately 
come  to  lie  between  N  —  t'  and  X  +  e,  no  matter  how  small  e  is.  Hence  the  x's 
approach  iV  as  a  limit.  Thus  under  either  supposition  the  suite  approaches  a  limit 
and  the  theorem  is  proved.  It  may  be  noted  that  under  the  second  supposition  tlie 
ic's  ultimately  lie  entirely  upon  one  side  of  tlie  x>oint  X  and  that  the  contlition 
|x„  +  ;,  —  a;„|  <  e  is  not  used  except  to  show  that  the  x"s  remain  finite. 

22.  Consider  next  a  set  of  points  (or  their  eorrelative  numbers) 
without  any  implication  that  tliey  form  a  suite,  that  is,  that  one  may 
be  said  to  be  subsequent  to  another.  If  there  is  only  a  finite  number 
of  points  in  the  set,  there  is  a  point  farthest  to  the  right  and  one 
farthest  to  the  left.  If  there  is  an  infinity  of  points  in  the  set,  two 
possibilities  arise.  Either  1°  it  is  not  possible  to  assign  a  point  A'  so 
far  to  the  right  that  no  point  of  the  set  is  farther  to  the  right  —  in 
which  case  the  set  is  said  to  be  unllinltcd  nhorc  —  or  2°  there  is  a 
point  K  such  that  no  point  of  the  set  is  lieyond  7v  —  and  the  set  is 
said  to  be  limited  above.  Similarly,  a  set  may  be  J  united  heJou-  or  ini- 
limited  heloiv.  If  a  set  is  limited  above  and  below  so  that  it  is  entirely 
contained  in  a  finite  interval,  it  is  said  merely  to  ])e  limited.  If  tliere 
is  a  point  C  such  that  in  any  interval,  no  matter  how  small,  surround- 
ing C  there  are  points  of  the  set,  then  C  is  called  a  paint  of  rnndt'nso- 
tion  of  the  set  (C  itself  may  or  may  not  belong  to  the  set). 

Theokem  4.  Any  infinite  set  of  points  which  is  limited  has  an 
upper  frontier  (maximum  ?),  a  lower  frontier  (minimum  ?),  and  at 
least  one  point  of  condensation. 

Before  proving  this  theorem,  consider  three  infinite  sets  as  illustrations : 
(a)   1,  1.9,  1.09,  1.999,  •  •  •.  (/3)   -  2,  •  •  •,  -  1.99,  -  1.9,  -  1, 

(7)   -l.-^.-i,  ■•■,1,  i,l. 

In  (a)  the  element  1  is  the  minimum  and  serves  also  as  the  lower  frontier  ;  it  is 
clearly  not  a  jxtint  of  condensation,  but  is  isolated.  There  is  no  maxinuun  ;  but  2 
is  the  upper  frontier  and  also  a  point  of  condensation.  In  (/3)  there  is  a  maxinuim 
—  1  and  a  minimum  —  2  (fur  —  2  has  been  incorporated  with  the  set).  In  (7)  there 
is  a  maxinuun  and  mininuim ;  the  point  of  condensation  is  0.  If  one  could  he  sure 
that  an  infinite  set  had  a  maxinuim  and  minimum,  as  is  the  case  witli  finite 
.sets,  there  would  be  no  need  of  considering  upper  and  lower  frontiers.  It  is  clear 
tliat  if  the  upju'r  or  lower  frontier  belongs  U)  the  set.  there  is  a  maximum  nr 
mininuun  and  the  frontier  is  not  necessarily  a  point  of  condensation  ;   whereas 


FUNDAMENTAL  THEORY-  39 

if  the  frontier  does  not  belong  to  the  set,  it  is  necessarily  a  point  of  condensation  and 
the  corresponding  extreme  j)oint  is  missing. 

To  prove  that  there  is  an  upi^er  frontier,  divide  the  points  of  the  line  into  two 
classes,  one  consisting  of  points  which  are  to  the  left  of  some  point  of  the  set,  the 
other  of  points  which  are  not  to  the  left  of  any  point  of  the  set  —  then  apply  the 
axiom.  Similarly  for  the  lower  frontier.  To  show  the  existence  of  a  point  of  con- 
densation, note  that  as  thei-e  is  an  infinity  of  elements  in  the  set,  any  point  p  is  such 
that  either  there  is  an  infinity  of  points  of  the  set  to  the  right  of  it  or  there  is  not. 
Hence  the  two  classes  into  which  all  points  are  to  be  assorted  are  suggested,  and 
the  application  of  the  axiom  offers  no  difficulty. 

EXERCISES 

1.  In  a  manner  analogous  to  the  proof  of  The<jrem  2,  show  that 

,    ,   , .      X  —  1       1  ,  ^x   , .      3  X  —  1       5  ,        , .       X-  +  1 

(a)  lim  =  -,  ((3)  Inn  =  -,  (7)     Inn    =— 1. 

a:  =  OX  —  2         2  x  =  2     X  +   5  7  .ti-lX'^  —  1 

2.  Given  an  infinite  series  .b'  =  Mi  +  »2  +  »3  +  •  •  •  .    Construct  the  suite 

.S'l  =  «i,    So  =  III  +  H2,    S3  =  III  +  "2  +  "3,    •  •  ••    Si  =  Ml  +i(2  +  •  •  ■  +  ",-.    •  •  -. 

where  Si  is  the  snnn  of  the  first  i  terms.  Show  that  Theorem  3  gives  :  The  neces- 
sary and  sufficient  t-Dudition  that  the  series  S  converge  is  that  it  is  possible  to  find 
an  n  so  large  that  j.Sn-i-;,  —  *'„  |  shall  be  less  than  an  assigned  e  for  all  values  of  p. 
It  is  to  be  understood  that  a  series  converges  when  the  suite  of  ,S'"s  approaches  a  limit, 
and  converse!}'. 

3.  If  in  a  series  Ui  —  Uo  +  W3  —  7(4  +  •  •  •  the  terms  approach  the  limit  0,  are 
alternately  positive  and  negative,  and  each  term  is  less  than  the  preceding,  the 
series  converges.    C<jnsider  the  suites  .Si.  .S3,  .S-.  •  •  •  and  ,S._,.  ,S4,  .Sq.  .... 

4.  Given  three  infinite  suites  of  numl.iers 

Xi.  Xo.  ■  ■  ■.  x„.  ■  ■  ■:  !/i.  i/-.  ■  ■  ■.  y„.  ■  ■  ■:  Zi.  z-2- ■  ■  ■■  Zn.  ■  ■  ■ 
of  which  the  first  never  decreases,  tlie  second  never  increases,  and  tlie  terms  of  tJie 
thiril  lie  between  corresponding  terms  of  tlie  first  two.  x,,  ^  Zn  s  (/„.  Siiow  that 
the  suite  of  2"s  has  a  xx.iint  of  condensation  at  or  between  the  limits  approached  by 
the  x's  and  liy  tiie  y's  ;  and  that  if  lim  x  =  lim  y  =  I.  then  the  z"s  api)i'oach  I  as  a 
limit. 

5.  Kestate  the  definitions  and  theorems  on  sets  of  points  in  arithmetic  terms. 

6.  Give  the  details  of  the  proof  of  Theorem  4.  Show  that  the  jirocjf  as  outlined 
gives  the  least  point  of  conden.sation.  How  would  the  proof  be  worded  so  as  to  give 
the  greatest  point  of  conden.sation?  Show  that  if  a  .-^et  is  limited  above, it  has  an 
upper  frontier  but  need  not  have  a  lower  frontier. 

7.  If  a  set  r)f  points  is  sucli  that  l)etween  any  two  there  is  a  third,  the  set  is  said 
to  be  dense.  Sliow  that  the  rationals  form  a  dense  .set  ;  also  the  irrationals.  Show 
that  any  point  of  a  dense  set  is  a  point  of  conden.sation  for  tlie  .set. 

8.  Show  that  the  rationals  p/(j  where  q  <  K  do  not  form  a  dense  set — in  fact 
are  a  finite  .set  in  any  limited  interval.  Hence  in  regarding  any  irrational  as  the 
limit  of  a  .set  of  rationals  it  is  necessary  that  the  denominators  and  also  the  numer- 
ators should  become  infinite. 


40  INTRODUCTORY  REVIEW 

9.  Show  that  if  an  infinite  set  (if  points  lies  in  a  limited  region  of  the  plane, 
say  in  the  rectangle  a  ^  x  ^  b.  c  ^  y  =  d,  there  must  be  at  least  one  point  of 
condensation  of  the  set.  Give  the  necessary  definitions  and  apply  the  axiom 
of  continuity  successively  to  the  abscissas  and  ordinates. 

23.  Real  functions  of  a  real  variable.  //  x  he  a  varlahle  which 
takes  on  a  certaiii  set  of  rallies  of  wlileJi  the  totality  may  he  denoted 
hy  [./']  and  if  y  is  a  second  variaJile  the  value  of  which  is  uniquely 
determined  for  each  x  of  tlie  set  [./■],  tJien  y  is  said  to  he  a  function  of 
X  defned  orer  the  set  [./•].  The  terms  "limited,"  "unlimited/'  "limited 
above,"  ''  unlimited  below,"  •  •  •  are  applied  to  a  function  if  they  are 
applicable  to  the  set  [//]  of  values  of  the  function.  Hence  Theorem  4 
has  the  corollary  : 

Theorem  5.  If  a  function  is  limited  over  the  set  [./•],  it  has  an 
upper  frontier  .1/  and  a  lower  frontier  m  for  that  set. 

If  the  function  takes  on  its  upper  frontier  J/,  that  is,  if  there  is  a 
value  x^  in  the  set  [./■]  suc-h  that  /(.'',-)  =  -V,  the  function  has  the  abso- 
lute viaximuiii  M  at  x^;  and  similarly  w"ith  respect  to  the  lower 
frontier.  In  any  case,  the  difference  M  —  iii  between  the  upper  and 
lower  frontiers  is  called  the  oscillation  of  the  function  for  the  set  [a-]. 
The  set  [;>•]  is  generally  an  interval. 

Consider  some  illustrations  of  functions  and  sets  over  which  they  are  defined. 
The  reciprocal  1/x  is  defined  for  all  values  of  x  save  0.  In  the  neighborhood  of  0 
the  function  is  unlimited  above  for  positive  x"s  and  unlimited  beluw  for  negative  x"s. 
It  should  be  noted  that  the  function  is  not  limited  in  the  interval  0  <  x  =  a  but  is 
limited  in  the  interval  e  ^  x  ^  a  where  e  is  any  assigned  positive  number.  The 
function  +  \'x  is  defined  fur  all  positive  x's  including  0  and  is  limited  below.  It 
is  not  limited  abitvc  for  the  totality  of  all  positive  numbers  ;  but  if  K  is  assigned, 
the  function  is  limited  in  the  interval  0  =  x  ^  A'.  The  factorial  function  x  I  is  de- 
fined only  for  positive  integers,  is  limited  beltjw  by  the  value  1,  but  is  not  limited 
above  unless  the  set  [x]  is  limited  above.  The  function  E  [x)  denoting  the  Integer 
not  greater  than  x  or  "tlie  integral  part  of  x"  is  defined  for  all  positivf  numbers 
—  for  instance  A"  (3)  =  E  (tt)  —  .3.  This  function  is  not  expressed,  like  the  elemen- 
tary functions  of  calculus,  as  a  "  fornuda  "  ;  it  is  defined  by  a  definite  law,  however, 
and  is  just  as  much  of  a  function  as  x-  -\-'?>x  -\-'l  or  \  sln-:ix  +  logx.  Indeed  it 
.should  be  noted  that  the  elementary  functions  themselves  are  in  the  first  instance 
defined  by  definite  laws  and  that  it  is  not  until  after  they  have  been  made 'the 
subject  of  considerable  study  and  have  lieen  largely  developeil  along  analytic  lines 
that  they  appear  as  formulas.  The  ideas  of  function  and  fonnula  are  essentially 
distinct  and  the  latter  is  essentially  secon<lary  to  the  former. 

The  definition  of  function  as  given  above  excludes  the  so-called  multiple-valued 
functions  such  as  \'x.  and  sin-i  x  where  to  a  given  value  of  x  correspond  more  than 
one  value  of  the  function.  It  is  usual,  however,  in  treating  nudtiple-valued  func- 
tions to  resolve  tlie  functions  into  different  parts  oi-  hranches  so  that  each  branch 
is  a  siuLile-valueil  function.  Thus  —  ^  x  is  one  brancii  and  —ax  the  other  branch 


FUXDAMEXTAL  THEORY  41 

of  vx  ;  in  fact  when  x  is  positive  tlie  symbol  V.c  is  usually  restricted  to  mean 
niei'ely  +  Vx  and  thus  bec(imes  a  single-valued  syndxil.  ( )ne  branch  of  sin-i  x  con- 
sists of  the  values  between  —  i  tt  and  +  4  tt,  otiier  branches  give  values  between 
^  TT  and  I  TT  or  —  Itt  and  —  |  tt,  and  so  on.  Hence  the  term  "function"  will  be 
restricted  in  this  chapter  to  the  single-valued  functions  allowed  by  the  definition. 

24.  If  X  =  (I,  Is  (inu  point  (if  (111  iiitercal  orcr  ir/tich  f(.i-)  is  dep'ned, 
the  f unci ioti  f(.r^  is  said  to  be  continuous  at  the  point  ./•  =  o  if 

lim/(./')  =fQi),  no  matter  hoir  x  =  a. 

X  =  a 

The  function  is  said  to  he  continnoi/s  in  the  intcrr(//  if  it  is  continuous 
at  every  pjoint  of  the  intercaJ.  If  the  function  is  not  continuous  at  the 
point  a,  it  is  said  to  Ije  disco?itinuous  at  ft ;  and  if  it  fails  to  be  con- 
tinuous at  any  one  point  of  an  interval,  it  is  said  to  be  discontinuous 
in  the  interval. 

Theorem  G.  If  any  finite  numl^er  of  functions  are  continuous  (at  a 
point  or  over  an  interval),  any  rational  expression  formed  of  those 
functions  is  continuous  (at  the  point  or  over  the  interval)  ])rovided  no 
division  by  zero  is  called  for. 

Theorem  7.  If  //  =_/'(,/•)  is  continuous  at  x^  and  takes  the  value 
y^=f(:r^  and  if  ,v  =  ^(//)  is  a  continuous  function  of  //  at  // =  v/^,  then 
z  =  <^  [/(•'')]  ^vill  be  a  continuous  function  of  x  at  ,/•,. 

In  regard  to  the  definition  of  continuity  note  that  a  funetiim  cannot  be  C(jn- 
tinuous  at  a  point  unless  it  is  defined  at  that  ixiint.  Thus  c-i/'"  is  not  continuous 
atx  =  0  because  division  by  0  is  impi^-^silde  and  the  fiuiction  is  umlefined.  If,  how- 
ever, the  function  be  defined  at  0  as/(0)  —  0,  the  functi(jn  becomes  continuous  at 
X  =  0.  In  like  manner  tlie  functicju  ]/x  is  not  continuous  at  the  (jrigin,  and  in  this 
case  it  is  impossible  to  assign  to/(0)  any  value  which  will  render  the  function 
continuous;  the  function  bec<jmes  infinite  at  the  origin  and  the  very  idea  of  be- 
coming infinite  precludes  the  possibility  of  approach  to  a  definite  Unfit.  Again,  the 
function  E  [x)  is  in  general  continuous.  Itut  is  discontinuous  for  integral  values 
of  X.  When  a  function  is  discontinuous  at  x  =  u.  the  amount  of  the  discontinuity  is 
the  limit  of  the  oscillation  ^f  —  m  of  tlie  function  in  the  interval  '(  —  5  <  x  <  «  +  5 
surrounding  the  X'oint  a  when  5  apijroaches  zero  as  its  liuut.  Tiie  discontinuity 
of  E  (x)  at  each  integral  value  of  x  is  clearly  1  ;  tliat  of  1/x  at  the  origin  is  infi- 
nite no  matter  what  value  is  assigned  ti)/(0). 

In  case  the  interval  over  which /(x)  is  defined  has  end  points,  say  «  =x  ^t), 
the  question  of  continuity  at  x  =  a  nuist  of  course  be  decided  by  allowing  x  to 
approach  a  from  the  right-hand  side  only  ;  and  similarly  it  is  a  (piestion  of  left- 
handed  approach  to  h.  In  general,  if  for  any  reason  it  is  desired  to  restrict  the 
approach  of  a  varialjle  to  its  linfit  to  being  one-sided,  tlie  notations  x  =  a+  and 
X  ==  h-  respectively  are  used  to  denote  approach  through  greater  values  (right- 
handed)  and  through  lesser  values  (left-handed).  It  is  ivit  necessary  to  make  this 
specification  in  the  case  of  tlie  ends  of  an  interval  ;  for  it  is  understood  that  x 
shall  take  on  only  values  in  the  interval  in  (luestioii.     It  should  be   noted  that 


42 


IXTRODUCTOllY  REVIEW 


lim  f{x)  =f(xo)  when  x  ==  Xo+  in  no  wise  implies  the  continuity  of  f{x)  at  Xq  ;  a 
simple  example  is  that  of  E  (x)  at  the  jiositive  integral  points. 

The  proof  of  Theorem  6  is  an  immediate  corollary  ajjplication  of  Theorem  2.    For 

lim  R  [/(x),  0  (x)  •  •  ■]  =  E  [lim/(x),  lim  0  (x),  •  •  •]  =  I^  [/(lii'i  ■^),  </>  (li"i  a;),  •  •  •], 
ajid  the  proof  of  Theorem  7  is  eciuallv  simple. 

Theokem  8.  If  /(■'')  is  continuous  at  ,/•  =  <>,  tlien  for  any  positive 
which  has  been  assigned,  no  matter  how  small,  there  may  be  found  a 
number  8  such  that  |/'(.r) —/'(*■/)  |  <  £  in  the  interval  j.r— (<|<S,  and 
hence  in  this  interval  the  oscillation  of  /(■'')  is  less  than  2  e.  And 
conversely,  if  these  conditions  hold,  the  function  is  continuous. 

\  This  theorem  is  in  reality  nothing  but  a  restatement  of  the  definition  of  conti- 
nuity combined  with  the  definition  of  a  limit.  For  "lini/(x)  =/(«)  when  x  =  a, 
no  matter  how"  means  that  the  difference  between /(x)  and/(«)  can  be  made  as 
small  as  desired  by  taking  x  sufficiently  near  to  a  ;  and  conversely.  The  reason 
for  this  restatement  is  that  the  present  form  is  more  amenable  to  analytic  opera- 
tions. It  also  suggests  the  geometric  picture  which  corre- 
sponds to  the  usual  idea  of  continuitj'  in  graphs.  For  the 
theorem  states  that  if  the  two  lines  y  =/(«)  ±  e  l>e  drawn, 
the  graph  of  the  function  remains  between  them  for  at  least 
the  short  distance  5  on  each  side  of  x  =  ((  ;  and  as  e  may  be 
assigned  a  value  as  small  as  desired,  the  graph  cannot  exhibit 
breaks.    On  the  other  hand  it  should  be  noted  that  the  actual 

physical  graph  is  not  a  curve  but  a  baud,  a  two-dimensional  region  of  greater  or 
less  breadth,  and  that  a  function  could  be  discontinuous  at  every  point  of  an 
interval  and  yet  lie  entirely  within  the  limits  of  any  given  physical  graph. 

It  is  clear  that  5,  which  has  to  be  detennined  subsaiuvntly  to  e,  is  in  general 
more  and  more  restricted  as  e  is  taken  smaller  and  that  for  different  points  it  is 
more  restricted  as  the  grai)h  rises  more  rapidly.  Tluis  if /(x)  =  1/x  an<l  e  =  1/1000, 
5  can  be  nearly  1/10  if  Xq  =  100.  Itut  must  be  sligiitly  less  tlian  1/1000  if  Xo  =  1,  and 
something  less  than  10^  'J  if  x  is  10-  '^.  Indeed,  if  x  be  allowed  to  approach  zero,  the 
value  5  for  any  assigned  e  also  approaches  zero  ;  and  although  the  function 
/(x)  =  1/x  is  continuous  in  the  interval  0  <  x  ^  1  and  for  any  given  Xq  and  e  a 
number  5  may  be  found  such  that  |/(x)  — /(xo)  ]  <  e  when  |x  —  Xoj  <  5,  yet  it  is  not 
possible  to  assign  a  number  5  which  shall  serve  uniformly  for  all  values  of  Xq. 

25.  TiiKoKKM  9.  If  a  function  /'(.'•)  is  continuous  in  an  interval 
'/  =i  ./•  s  //  witli  end  points,  it  is  jiossiblt^  to  find  a  8  such  that 
/'(./•) —/'(./•u)  <€  when  j.''  — .'■oj<8  for  all  ])oints  .ro ;  and  the  function 
is  said  to  be  unifiiniil y  couft/ii/oi/s. 

The  ])r<i()f  is  conducted  by  the  method  of  reductio  ad  absurdum.  Suppose  e 
is  assigned.  Consider  the  suite  of  values  ',.  |.  ',.  •••.  or  any  other  suite  which 
approaches  zer<i  as  a  limit.  Su])pose  that  no  one  of  these  values  will  serve  as  a  5 
for  all  points  of  the  interval.  Then  there  niu^t  be  at  least  one  point  for  which  .] 
will  not  sei-ve.  at  least  one  for  wliicli  ]  will  not  serve,  at  least  one  for  which  J  will 
not  serve,  and  so  on  indefinitely.    This  infniite  set  of  points  nmst  have  at  least  one 


FUXDA^rEXTAL  THEORY  43 

point  of  condensation  C  such  tiiat  in  any  interval  surroundinf;  C  tliere  are  points  for 
which  2-^  will  not  serve  as  5,  no  matter  how  large  k.  But  now  by  hypothesis /(x) 
is  continuous  at  C  and  hence  a  number  5  can  be  found  such  that  |/(x)  — /(C')|<  I  e 
when  \x  —  Xo\<  2S.  The  oscillation  of  f{x)  in  the  whole  interval  4 3  is  less  than  e. 
Now  if  xo  be  any  point  in  the  middle  half  of  this  interval,  |xo—  C  \  <  S  ;  and  if  x 
satisfies  the  relation  \x  —  Xo]  <  S,  it  must  still  lie  in  the  interval  4  5  and  the  differ- 
ence \f{x)  — /(xo)  I  <  e,  being  surely  not  greater  than  the  oscillation  of  /in  the  wdiole 
interval.  Hence  it  is  possible  to  surround  C  with  an  interval  so  small  that  the 
same  5  will  serve  for  any  point  of  the  interval.  This  contradicts  the  former  con- 
clusion, and  hence  the  hj'pothesis  upon  wliich  that  conclusion  was  based  nuist  have 
been  false  and  it  must  have  been  possible  to  find  a  5  which  would  serve  for  all 
points  of  the  interval.  The  reason  why  the  proof  would  not  apply  t(»  a  function 
like  l/x  defined  in  the  interval  0  <  x  s  i  lacking  an  end  point  is  precisely  that 
tlie  point  of  condensation  C  Avould  be  0,  and  at  0  the  function  is  not  continuous 
and  [/(x)  —f{C)  |  <  i  e,  jx  —  C  |  <  2  5  could  not  be  satistied. 

Theokem  10.  If  a  function  is  continuous  in  a  region  which  inchides 
its  end  points,  the  function  is  limited. 

Theorem  11.  If  a  function  is  continuous  in  an  interval  which  includes 
its  end  points,  the  function  takes  on  its  upper  frontier  and  has  a  niaxi- 
luuni  JI;  similarly  it  has  a  minimum  ))/. 

These  are  successive  corollaries  of  Theorem  9.  For  let  e  be  assigned  and  let  5 
be  determined  so  as  to  serve  uniformly  fnr  all  points  of  tlic  interval.  IJivide  the 
interval  b  —  a  into  ?i  successive  Intervals  of  length  5  or  less.  Tlieii  in  each  such 
interval /cannot  increase  by  more  than  e  nor  decrease  by  more  than  e.  Hence  / 
will  be  contained  betweeii  the  values/(</)  +  ne  and /(a)  —  ne.  and  is  limited.  And 
fix)  has  an  tipper  and  a  lower  frontier  in  tlie  interval.  Next  consider  the  rational 
function  1/(3/  —  /)  of/.  By  Theorem  0  this  is  continuous  in  the  interval  tuiless 
the  denominator  vanishes,  and  if  contiimous  it  is  limited.  Tliis,  Jiowcver.  is  impos- 
sible for  the  reason  that,  as  3/  is  a  frontier  of  values  of  /.  the  difference  3f  —  / 
may  be  made  as  small  as  desired.  Hence  l/{M  —  f)  is  not  continuous  and  there 
nuist  be  some  value  of  x  for  which /=  M. 

Theorem  12.  li  f(:r)  is  continuous  in  the  interval  >'  ^  ./'^  A  Avith  end 
])()ints  and  if /'(")  and /(/y)  have  o})posite  signs,  there  is  at  least  one 
])oint  $,"<$<  //,  in  the  interval  for  which  the  function  vanislies. 
And  wh(4her  f (V/)  and_/(A)  have  o})posite  signs  oi'  not,  there  is  a  point 
$,"<.$<.  ft,  such  that /'(I)  =  /M,  Avhere  /x,  is  any  value  intermediate  be- 
tween the  maximum  and  minimum  of  /'  in  the  interval. 

For  convenience  suppose  tliat/(//)  <  0.  Then  in  the  neighborhood  of  x  =  (i  the 
function  will  remain  negative  on  accouiit  of  its  continuity  ;  and  in  the  neighbor- 
hood of  h  it  will  remain  positive.  Let  ^  be  the  lower  frontier  of  values  of  x  which 
make/(x)  positive.  Suppose  that/(^)  were  either  positive  or  negative.  Then  as 
/  is  continuous,  an  interval  could  be  chosen  surrounding  ^  and  so  small  that  /  re- 
mained positive  or  negative  in  that  interval.  In  neither  case  could  ^  be  the  lower 
frontier  of  positive  values.    Hence  the  contradiction,  and  /(|)  must  be  zero.    To 


44  INTRODUCTORY  REVIEW 

prove  the  second  part  of  the  theorem,  let  c  ajid  d  be  the  vahies  of  x  which  make 
/  a  minimum  and  maxinuim.  Then  the  function  f  —  n  has  opposite  signs  at  c  and 
d,  and  must  vanisli  at  some  point  of  the  interval  between  c  and  d  ;  and  hence  a 
fortiori  at  some  point  of  the  interval  from  a  to  b. 

EXERCISES 

1.  Note  that  x  is  a  continuous  function  of  x,  and  that  consecpiently  it  follows 
from  Theorem  6  that  any  rational  fraction  P(x)/Q(x),  where  /'  and  Q  are  poly- 
nomials in  X,  must  be  continuous  for  all  x"s  except  roots  of  Q{x)  =  0. 

2.  Graph  the  function  x  —  E  (x)  for  x  ^  0  and  show  that  it  is  continuous  except 
for  integral  values  of  x.  Show  that  it  is  limited,  has  a  minimum  0,  an  upper  fron- 
tier 1,  but  no  maxinuim. 

3.  Suppose  that/(x)  is  defined  for  an  inlinite  set  [xj  of  which  x  =  a  is  a  point 
of  condensation  (not  necessarily  itself  a  point  of  the  set).    Suppose 

lim     [/(x')-/(x")]  =0     or     |/(x') -/(x")  |<  e,  |x' -  «|  <  5,  |x"  -  « |<  5, 

when  x'  and  x"  regarded  as  independent  variables  approach  a  as  a  limit  (passing 
only  over  values  of  the  set  [x],  of  course).  Show  that /(x)  approaches  a  limit  as 
X  =  a.  By  considering  the  set  of  values  of /(x),  the  method  of  Theorem  3  applies 
almost  verbatim.  Show  that  there  is  no  essential  change  in  the  proof  if  it  be 
assumed  that  x'  and  x"  become  infinite,  the  set  [x]  being  unlimited  instead  of 
having  a  jioint  of  condensation  a. 

4.  From  the  formula  sin  x  <  x  and  the  formulas  for  sin  u  —  sin  v  and  cos  u  —  cos  u 
show  that  A  sin  X  and  A  cos  x  are  numerically  less  than  2 1  Ax  ] ;  hence  infer  that  sin  x 
and  cosx  are  continuous  functions  of  ,c  for  all  values  of  x. 

5.  Wliat  are  the  intervals  of  continuity  f(U'  tan.c  and  esc  x  ?  If  e  =  lO-**,  what 
are  approximately  the  largest  available  values  of  5  that  will  make  |/(x)  — /(x^)  |<£ 
when  Xj,  =  1°,  30^,  (50°,  SU'^  for  each  ?    Use  a  four-place  table. 

6.  Let  /(x)  b(!  defined  in  tlie  interval  from  0  to  1  as  (Mjual  to  0  when  x  is  irra- 
tional and  e(iual  to  ]/(/  when  x  is  rational  ami  expressed  as  a  fraction  p/q  in  lowest 
terms.  SIiow  that/  is  continuous  fur  irrational  valiU's  and  discontinuous  for 
rational  values.    Ex.  8,  p.  39,  will  be  of  assistance  in  treating  the  irrational  values. 

7.  Note  that  in  the  definition  of  contiiuiity  a  generalization  may  be  introduced 
by  allowing  the  set  [x]  over  which/  is  defined  to  be  any  set  each  point  of  which 
is  a  point  of  condensation  of  the  set,  and  that  hence  continuity  over  a  dense  set 
(Ex.  7  above),  say  tlie  rationals  or  irrationals,  maj'  be  defined.  This  is  important 
because  many  functions  ar(_^  in  tlie  first  instance  defined  only  for  rationals  and  are 
subse(iuently  defined  for  iri'ationals  by  interpolation.  Note  that  if  a  function  is 
contimious  over  a  dense  set  (say,  the  rationals),  it  docs  not  follow  tliat  it  is  uni- 
formly contiimous  over  the  si't.  For  the  point  of  condensation  C  which  was  used 
in  the  proof  of  Theorem  !)  may  not  be  a  point  of  t!ie  set  (may  be  irrational),  and 
the  proof  wouhl  fall  tlirougli  for  tlie  same  reason  tliat  it  would  in  the  case  of  1/x 
in  the  interval  0  <  x  ^  1.  namely.  l.)eea\ise  it  could  not  be  afiirme(l  that  the  function 
was  contimious  at  ('.  Show  that  if  a  func'tiou  is  defined  and  is  uniformly  contimi- 
ous over  a  dense  set,  the  value /(/•)  will  apjiroach  a  limit  wlieii  x  approaches  any 
value  a   (not  necessarily  of  th(!  set,  but  situated  between  the  upper  and  lower 


FUNDAMENTAL  THEORY  45 

frontiers  of  the  set),  and  that  if  this  limit  be  defined  as  the  value  of  /(a),  tlie 
function  will  remain  continuous.    Ex.  3  may  be  used  to  advantage. 

8.  By  factoring  (x  +  A  x)"  —  x",  show  for  integral  values  of  n  that  when 
OS  a;  ^  A",  then  A  (x")  <?i7v»-i  Ax  for  small  Ax's  and  consequently  x"  is  uniformly 
continuous  in  the  interval  0^  x^  K.  If  it  be  assumed  that  x"  has  been  defined 
only  for  rational  x's,  it  follows  from  Ex.  7  that  the  definition  may  be  extended 
to  all  x's  and  that  the  resulting  x"  will  be  continuous. 

9.  Suppose  (a)  tliat/(x)  +f{y)  =/(x  +  y)  for  any  numbers  x  and  y.  Show  that 
f{n)  =  n/(l)  and  nf{l/n)  =/(l),  and  hence  infer  that  f{x)  =  x/(l)  =  Cx,  where 
C=/(l),  for  all  rational  x's.  From  Ex.  7  it  follows  that  if  /(x)  is  continuous, 
/(x)  =  Cx  for  all  x's.  Consider  (p)  the  function /(x)  such  that/(x)/(?/)  =/(x  +  ?/). 
Show  that  it  is  Ce^  =  a^\ 

10.  Show  by  Theorem  12  that  if  y  —f{x)  is  a  continuous  constantly  increasing 
function  in  the  interval  a  ^  x  ^  f)^  then  to  each  value  of  y  corresponds  a  single  value 
of  X  so  that  the  function  x  =f~^  (y)  exists  and  is  single- valued  ;  show  also  that 
it  is  continuous  and  constantly  increasing.  State  the  corresponding  theorem  if 
/(x)  is  constantly  decreasing.  The  function  f-'^(y)  is  called  the  inverse  function 
to/(x). 

11.  Apply  Ex.  10  to  discuss  y  =  Vx,  where  n  is  integral,  x  is  positive,  and  only 
positive  roots  are  taken  into  consideration. 

12.  In  aritlnnetic  it  may  readily  be  shown  that  the  equations 

a"^/"  —  a"'  +  «,  (a'")"  =  «"'»,  a"h"  =  (ah)", 

are  true  when  a  and  b  are  rational  and  positive  and  when  m  and  n  are  any  positive 
and  negative  integers  or  zero,  (a)  Can  it  be  inferred  that  they  hold  when  a 
and  b  are  positive  irrationals  ?  (/3)  How  about  the  extension  of  the  fundamental 
inequalities 

x">l,     when     x  >  1,  x"  <  1,     when     OSx<l 

to  all  rational  values  of  n  and  the  proof  of  the  ineiiualities 

x"'>x'«     if     m>n     and     x>l,  x"'<x'»     if     vi>n     and     0<x<l. 

(7)  Next  consider  x  as  held  constant  and  the  exponent  n  as  variable.  Discuss  the 
exponential  function  a^  from  this  relation,  and  Exs.  10,  11,  and  other  theorems  that 
may  seem  necessary.    Treat  the  logarithm  as  the  inverse  of  the  exponential. 

26.  The  derivative.  Jf  x  =  c  is  a  point  of  "n  Interval  over  which 
f{x)  is  defined  and  if  the.  ([Kotient 

A/'        f(a  +  h)  -f(a) 

-—  = ,         h  =  A,r, 

approaches  a,  limit  ivlien  h  aptpn'oaehes  zero,  no  matter  hoiv,  the  function 
f(x')  is  said  to  he  differentiahle  at  x  =  a  and  the  rahce  of  the  limit  of 
the  quotient  is  the  derivative  f '(a)  of  f  at  x  =  a.  In  the  case  of  differ- 
entiability, the  definition  of  a  limit  gives 

■^^"^^'l"-^^"^  =f(")+ri      or     f{a-{-h)-f{a)  =  hf\a)+r^h,      (1) 
where  liiu  r;  =  0  when  lini  h  =  0,  no  matter  how. 


46  IXTRODUCTORY   KEVIEW 

In  other  words  if  e  is  piven,  a  5  can  be  found  so  that|r;|<e  when  \h\<S.  This 
shows  that  a  function  differentiable  at  a  as  in  (1)  is  continuous  at  a.    For 

|/(a  +  h)-f{a)\^\r{a)\8+  e8,  \h\<  5. 

If  the  limit  of  the  quotient  exists  when  h  =  0  through  positive  values  only,  the 
function  has  a  right-hand  derivative  which  may  be  denoted  by/'  {a+)  and  similarly 
for  the  left-hand  derivative /'(a-).  At  the  end  points  of  an  interval  the  derivative 
is  always  considered  as  one-handed  ;  but  for  interior  points  the  right-hand  and  left- 
hand  derivatives  must  be  equal  if  the  function  is  to  have  a  derivative  (unqualified). 
The  function  is  .said  to  have  an  infinite  derivative  at  a  if  the  quotient  becomes  infi- 
nite as  7i  =  0  ;  but  if  a  is  an  interior  point,  the  quotient  must  become  positively 
infinite  or  negatively  infinite  for  all  manners  of  approach  and  not  positively  infinite 
for  some  and  negatively  infinite  for  others.  Geometrically  this  allows  a  vertical 
tangent  with  an  inflection  point,  but  not  with  a  cusp  as  in  Fig.  3,  p.  8.  If  infinite 
derivatives  are  allowed,  the  function  maj'  have  a  derivative  and  yet  be  discontin- 
uous, as  is  suggested  hj  any  figure  where /(a)  is  any  value  between  lim/(x)  when 
X  =  a+  and  lim/(x)  when  x  =  «-. 

Theorem  13.  If  a  function  takes  on  its  maximum  (or  minimum)  at 
an  interior  point  of  the  interval  of  deiinition  and  if  it  is  differentiable 
at  tliat  point,  the  derivative  is  zero. 

Theorem  14.  Boilers  TJieorem.  If  a  function  /'(,'■)  is  continuous  over 
an  interval  "  =  .'■  =  h  with  end  jjoints  and  vanishes  at  the  ends  and  has 
a  derivative  at  each  interior  point  -:'  <  ./•  <  //,  there  is  some  }ioint  L 
a  <  ^  <  h,  such  that  f  ($)  =  0. 

Theorem  15.  Tlicnri'm  of  flte  Jfran.  If  a  function  is  continuous  over 
an  interval  "  ^  .'•  ^  />  and  has  a  derivative  at  each  interior  point,  there 
is  some  point  i  such  that 

■AM -/(.- 1  ^  .,,       „,,    /'-'  +  ^'-/'"'=,v-  +  OA), 

h  —  (I  h  ' 

where  h  ^  I,  —  a*  and  6  is  a  proper  fraction,  0  <  ^  <  1. 

To  prove  the  first  theorem,  note  that  if /(a)  =  ,V,  the  dift'erence/(a  -{-  h)  —  f{a) 
cannot  be  positive  for  any  value  of  h  and  the  quotient  Af/h  cannot  he  positive 
when  /i  >  0  and  cannot  be  negative  when  /t<0.  Hence  the  right-hand  derivative 
cannot  be  positive  and  the  left-hand  derivative  cannot  be  negative.  As  these  two 
must  be  equal  if  the  fuiK^tion  has  a  derivative,  it  follows  tliat  tlicv  nuist  be  zero, 
and  the  derivative  is  zcrn.  Tiie  secimd  theorem  is  an  immciliate  cnrnliary.  Fur  as 
the  function  is  continuous  it  nuist  have  a  niaxinmm  and  a  mininiuni  (Tlicorcm  11) 
both  of  which  cannot  be  zero  unless  the  function  is  always  zero  in  the  innT\al. 
Now  if  the  function  is  identically  zei'o.  the  derivative  is  ideiuically  zero  and  the 
theorem  is  true  ;  whereas  if  the  function  is  not  identically  zero,  either  the  niaxiuunu 
or  minimum  nnist  be  at  an  interior  ]ioint.  and  at  that  point  t  lie  derivative  will  vauisii. 

*  That  the  tlicnrcni  is  true  for  any  part  of  tlic  interval  from  n  to  h  if  it  is  ti'uc  for  tlie 
whole  interval  follows  from  the  fact  that  the  eonditioiis.  namely,  that  J'  be  eoiitinuous 
and  that/''  exist,  lioM  for  any  part  of  tlie  int-rval  if  they  hoM  for  the  wlioje. 


FUXDAMEXTAL  THEOKY  47 

To  prove  the  last  theorem  construct  the  auxiliary  function 

h —a  0  —  a 

As  \j/  [a)  =  yp  (b)  =  0,  Holle's  Theorem  shows  that  there  is  some  point  for  which 
\t'(^)  =  0,  and  if  this  value  be  substituted  in  the  expression  for  ^'(x)  the  solution 
f"i"/'(l)  gives  the  result  demanded  by  the  theorem.  The  proof,  however,  requires 
the  use  of  the  function  ^  (j)  and  its  derivative  and  is  not  complete  until  it  is  shown 
that  \f/  (x)  really  satisfies  the  conditions  of  Rolle's  Theorem,  namely,  is  continuous 
in  the  interval  a^x^b  and  has  a  derivative  for  every  point  a  <x<  h.  The  con- 
tinuity is  a  consequence  of  Theorem  6  ;  that  the  derivative  exists  follows  from  the 
ilirect  application  of  the  definition  combined  with  the  assumption  that  the  deriva- 
tive of /exists. 

27.  TiiEOEKM  Ki.  If  a  fuiu-tion  lias  a  derivative  which  is  identically 
zero  ill  the  interval  "  =  ./■  =  b,  the  fnnction  is  constant ;  and  if  two 
functions  have  deri\atives  e(|ual  throughout  the  interval,  tlie  functions 
differ  by  a  constant. 

Theoke.vi  17.  If  /(•'•)  is  differential )le  and  l)econies  infinite  when 
X  ==  a,  the  derivative  cannot  remain  finite  as  ./•  =  a. 

Theorem  18.  If  the  derivative  /'(■'')  of  a  function  exists  and  is  a 
continuous  function  of  .'•  in  the  interval  "  =  .'■  ^  h,  the  quotient  \f/h 
converges  uniformly  toward  its  limit /'(./•). 

These  theorems  are  consequences  of  the  Theorem  of  the  Mean.    For  the  first, 

/(«  +  h)-f{a)  =  A/'('f  +  eh)  =  0.     if     h^b-  a,     or    /(«  +  Ji)  =/(«). 

Hence /(j)  is  constant.  And  in  case  of  two  functions/and  0  with  equal  derivatives, 
the  difference  xp  (x)  =f(x)  —  4>(x)  will  have  a  derivative  that  is  zero  and  the  differ- 
ence will  be  constant.  For  the  second,  let  x,,  be  a  fixed  value  near  a  and  suppose  that 
in  the  interval  from  x^  to  a  the  derivative  remained  finite,  say  less  than  K.    Then 

I  /(xo  +  l<)  -  f{xo)  I  =  I  hr{xo  +  eh)\^\h\K. 

Now  let  Xq  +  h  approach  a  and  note  that  the  left-hand  term  becomes  infinite  and 
the  supposition  that/'  remained  finite  is  contradicted.  For  the  third,  note  that/', 
being  continuous,  must  be  unifnrmly  continuous  (Theorem  9),  and  lience  that  if  e  is 
given,  a  5  may  be  found  surh  that 

\l^^:±Ilz:m-f'ix)\^\r(x  +  eh)-r{x)\<e 


when  |/ii<  5  and  for  all  .c's  in  the  interval  ;  and  the  theorem  is  proved. 

Concerning  derivatives  of  higher  order  no  special  remarks  are  necessary.  ICach 
is  the  derivative  of  a  definite  function  —  the  previous  derivative.  If  the  deri\a- 
tives  of  the  first  ?i  orders  exist  and  are  continuous,  the  derivative  of  order  n  +  1 
ma}"  or  may  not  exist.  In  practical  aiiplications,  however,  tlie  functions  are  gen- 
erally indefinitely  differentiable  except  at  certain  isolated  points.  The  pri.mf  of 
LeibmV/s  Theorem  (S  8)  may  be  revised  so  as  to  depend  mi  elementary  processes. 
Let  the  f(n-mula  lie  assumed  for  a  given  value  of  n.    The  only  terms  which  can 


48  INTKODUCTOKY   KEVIEW 

contribute  to  the  term  I)HiD"  +  ^-'v  in  the  formula  for  the  (?i  +  l)st  derivative  of 
uv  are  the  terms 

n(n— I)  ■  ■  ■  (n  — i  +  2)  ^^.    ,     r,    .,    ■  n(n  —  1)  ■  ■  •  (n  —  i  +  1)  -r,-    ^^ 

1.2...(t-l)  1.2...t 

in  wliicli  tlie  first  factor  is  to  be  differentiated  in  the  first  and  the  second  in  tlie 
second.    Tlie  sum  of  tlie  coelticients  obtained  by  differentiating  is 

71  (>i  -  1)  •  ■  •  {n  -  i  +  2)       n{n-  ])  ■  •  ■  {n  -  i  +  1)  _  (n  +  l)n  •  ■  ■  (n-  i  +  2) 
1  ■  2  •  •  •  (i  -  1^  1  •  2  •  •  ■  j  "  l-2---i  ' 

which  is  precisely  the  proper  coefficient  for  the  term  D'uD'^  +  i  -  'i-  in  the  expansion 
of  the  {n  +  1)  St  derivative  of  uv  by  Leibniz's  Theorem. 

With  regard  to  this  rule  and  the  otlier  elementary  rules  of  operation  (4)-(7)  of 
the  previous  chapter  it  should  be  remarked  that  a  theorem  as  well  as  a  rule  is  in- 
volved—  thus:  If  two  functions  u  and  v  are  differentiable  at  .r„,  then  the  product 
uv  is  differentiable  at  x^,  and  the  value  of  the  derivative  is  u  (jv)  v'  {x^^)  +  u'  (x^  v  (x^). 
And  similar  theorems  arise  in  connection  witli  the  other  rules.  As  a  matter  of  fact 
the  ordinary  proof  needs  only  to  be  gone  over  with  care  in  order  to  convert  it  into 
a  rigorous  demonstration.  But  care  does  need  to  be  exercised  both  in  stating  the 
theorem  and  in  looking  to  the  proof.  For  instance,  the  aliove  theorem  concerning 
a  product  is  not  true  if  infinite  derivatives  are  allowed.  For  let  it  be  —  1,  0,  or  +  1 
according  as  x  is  negative,  0,  or  positive,  and  let  v  =  x.  Now  v  has  always  a  deriva- 
tive which  is  1  and  u  has  always  a  derivative  which  is  0,  +  co,  or  0  according  as  x 
is  negative,  0,  tir  positive.  The  product  uv  is  |x|,  of  which  the  derivative  is  —  1  for 
negative  x's,  +1  for  positive  x's,  and  nonexistent  for  0.  Here  the  product  has  no 
derivative  at  0,  although  each  factor  has  a  derivative,  and  it  would  be  useless  to  have 
a  formula  for  attempting  to  evaluate  something  that  did  not  exist. 


EXERCISES 

1.  Show  that  if  at  a  point  tlie  deiivative  of  a  function  exists  and  is  positive,  the 
function  must  be  increasing  at  that  point. 

2.  Suppose  that  the  derivatives  /'(«)  and  fij))  exist  and  are  not  zero.  Show 
that /(a)  and  fQj)  are  relative  maxima  or  minima  of  /  in  the  interval  a  =  x  ^  ?>,  and 
determine  the  precise  criteria  in  terms  of  the  signs  of  the  derivatives /'(«)  and /'(';). 

3.  Show  that  if  a  contimious  function  has  a  positive  right-hand  derivative  at 
every  point  of  the  interval  a^x'^h.  then /(?>)  is  the  maxinuim  value  of/.  Simi- 
larly, if  the  right-hand  derivative  is  negative,  show  tliat/(^)  is  the  mlninuim  of/. 

4.  Apply  the  Theorem  of  the  Mean  to  show  that  if /'(.r)  is  continuous  at  a,  then 

x\  .'"  ==  <i         X    —  X 

x'  and  x"  lieing  regarded  as  independent. 

5.  Form  the  increments  of  a  function /for  cquicrcscent  values  of  the  variable  : 

A  J  =  f(a  +  h)  -  fin).  XJ  =  f(<t  +  2  //)  -  f{a  +  A), 

A,/  =  /(a. +  a/o-/("  +  -^')----- 


FUNDAMENTAL  THEORY  49 

These  are  called  first  differences  ;  the  differences  of  these  differences  are 
Af/  =  /(r/  +  2/0  -  2f{a  +  h)+f{a), 
Asf  =f{a  +  3  h)  -  2/(a  +  2  /;)  +  f{a  +  h),  ■  ■  ■ 
which  are  called  the  second  differences  ;  in  like  manner  there  are  third  differences 

Af/  =/(r/  +  3  h)  -  Sf{a  +  2h)  +  8f{a  +  h)-f{n),  ■  ■  ■ 
and  so  on.    Apply  the  Law  of  the  3Iean  to  all  the  differences  and  show  that 
Ai-/=  h-f"{a  +  d^h  +  e.Ji),         A^.f=  hV"{a  +  O^h  +  OJi  +  <9J0,  •  •  ■  • 
Hence  show  that  if  the  first  n  derivatives  of /are  continuous  at  a,  then 

A  =  0    ft-  !i  =  Q    ft"  h  =  n    ft" 

6.   Cnuchy's   Theorem.    If /(j)   and  <p(x)   are  contiiuious  over  r/ ^  j  ^ /<,  have 
derivatives  at  each  interior  point,  and  if  0'(.'")  does  not  vanish  in  the  interval, 

/(6)_/(r/)  _/'(t)     ^^      f(a  +  h)-fia)  _  .f{a  +  eh) 


<t>  (6)  -  4>  {(I)      (t>'(k)  4>{ei  +  h)-  cf, (a)      <t>'{a  +  eii) 

Prove  that  this  follows  from  the  application  of  Kolle's  Theorem  to  the  function 

^(x)  =f{x)-f{a)  -  [0(x)  -  0(«)]  ff;^~{f| . 

7.  One  application  of  Ex.  H  is  to  the  theory  of  indeterminate  forms.  Show  that 
if /(a)  =  (p(ri)  =  0  and  if /'(.r)/(/)'(.r)  approaches  a  limit  when  x  =  «,  W\(i\\  f  (/)  /  <p  {x) 
will  approach  the  same  limit. 

8.  Taylor's  Theorem.  Note  that  the  form  f{b)  —f{n)  +  {h  —  it)f'{^)  is  one  way 
of  writing  the  Theorem  of  the  ^lean.    By  the  application  of  ]{olie"s  Theorem  to 

f{h)-f{a)-(h-a)f'{a) 


V  (x)  =f{h)-f{x)  -  (b  -  x)f'{x)  -  (h  -  .c)-  ■ 


{h  -  «)- 


show  f{h)  =f{,i)  +  {h  -  n)f'('i)  +  .<''--^^/"(t), 

and  to   i  (X)  =  f{b)  -  fix)  -  {h  -  x)r{x)  -  l^^/-(.f ) lJ^'')'ll'/(..  -D  (x) 

-  TT^F-^'^'')  -•^^")  -  ^''  -  ")-^'(") 
{b  -  «)"  L 

_  ^^.f"(a) L 1       J  («-.,(„)    , 

2  {n—l)\  J 

show  f{li)  ^f{a)  +  {b  -  a) /'{<,)  +  ^^^-f"{a)  +  •  • . 

{n  —  1)  !  n\ 

"What  are  the  restrictions  that  must  be  imposed  on  the  function  and  its  derivatives  '? 

9.  If  a  continuous  functi<m  over  a  ^x  =  b  has  a  riirht-hand  derivative  at  each 
point  of  the  interval  whicli  is  zero,  show  that  the  function  is  constant.  .Vpply  Ex.  2 
to  the  f  unctions /(x)  +  e  (j- —  r/)  and/(x)  —  £(j  —  r()  to  show  that  the  maximum 
difference  between  the  functions  is  2e{b—  a)  and  that/  nuist  tlierefore  be  constant. 


50 


INTRODUCTORY   REVIEW 


10.  State  and  prove  the  theorems  implied  in  the  formulas  (4)-(6),  p.  2. 

11.  Consider  the  extension  of  Ex.  7,  p.  44,  to  derivatives  of  functions  defined 
over  a  dense  set.  If  the  derivative  exists  and  is  uniformly  continuous  over  the  dense 
set,  what  of  the  existence  and  continuity  of  the  derivative  of  the  function  when  its 
definition  is  extended  as  there  indicated  ? 

12.  If  f(x)  has  a  finite  derivative  at  each  point  of  the  interval  a^x^h,  the 
derivative  f'{x)  must  take  on  evei-y  value  intermediate  between  any  two  of  its  values. 
To  show  this,  take  first  the  case  where /'(a)  and/' (6)  have  opposite  signs  and  show, 
by  the  continuity  of  /  and  by  Theorem  13  and  Ex.  2,  that  /'(f)  =  0.  Next  if 
/'(a)  <  M  <./" '('')  without  any  restrictions  on /'(a)  and /'(/>),  consider  the  function 
/(x)  — /xx  and  its  derivative  /'(x)  — /x.  Finally,  prove  the  complete  theorem.  It 
should  be  noted  that  the  continuity  of  /'(x)  is  not  assumed,  nor  is  it  proved ;  for 
there  are  functions  which  take  every  value  intermediate  between  two  given  values 
and  yet  are  not  continuous. 

28.  Summation  and  integration.  Let  /(.r)  be  defined  and  limited 
over  the  interval  o  ^  j-  ^  b  and  let  M,  m,  and  0  =  M—  m  be  the 
upper  frontier,  lower  fron- 
tier, and  oscillation  of  /(•'■) 
in  the  interval.  Let  ?i  —  1 
points  of  division  l)e  intro- 
duced in  the  interval  divid- 
ing it  into  71  consecutive 
intervals  S^,  So,  •••.  S„  of 
which  the  largest  has  the 
length  A  and  let  .1/,-,  ///,-,  O., 

and  f(i,)  he  the  up})er  and  lower  frontiers,  the  oscillation,  and  any 
value  of  the  function  in  the  interval  8,-.    Then  the  inequalities 

will  hold,  and  if  these  terms  be  summed  up  for  all  ?i  intervals, 


Y 

3/; 

^l 

/ 

/ 

in 

0 

c 

c 

'i 

I 

>         X 

m{h  -  a)  ^  2^/y;A  ^  2)/Yv^-)8,  ^  2)-l/A-  ^  ^^ij' 


') 


(^•0 


wall  also  hold.  Let  .^  =  2//^8,,  o-  =  2/'(^,)8,-,  and  ,s'  =  SJ/^.  i'l'^'n^  (-0 
it  is  clear  that  the  difference  .s'  —  .v  does  not  exceed 

(J/  -  ///)  (h  -  <i)  =  0(J>  -  n), 

the  product  of  the  length  of  the  interval  by  the  oscillation  in  it.  The 
values  of  the  sums  S,  s,  cr  Avill  evidently  depend  on  the  numlx'r  f)f  ]iarts 
into  which  the  interval  is  divided  and  on  the  Avay  in  which  it  is  dividiMl 
into  that  nuniber  of  ])arts. 

Thkoi;e:\i  19.   If  n'  additional  ]i(»ints  of  division  be  introduced  into 
the  interval,  the  sum  .S'  constructed  for  the  'a  4-  />'  —  1  points  <-)f  division 


FUNDAMENTAL  THEOKY  51 

cannot  be  greater  than  S  and  cannot  be  less  than  .S'  by  more  than 
7i'(>A.  Similarly,  *•'  cannot  be  less  than  s  and  cannot  exceed  s  by  more 
than  7i'(jA. 

Theokem  20.  There  exists  a  lower  frontier  L  for  all  possible  methods 
of  constructing  the  sum  S  and  an  upper  frontier  I  for  s. 

Theokem  21.  Darboux's  Tlteoreiii.  When  e  is  assigned  it  is  possible 
to  find  a  A  so  small  that  for  all  methods  of  division  for  wliich  8,-  ^  A, 
the  sums  .S'  and  .v  shall  differ  from  their  frontier  values  L  and  I  by  less 
than  any  preassigned  e. 

To  prove  the  first  theorem  note  that  although  {A)  is  written  for  the  whole  inter- 
val from  a  to  h  and  for  the  svuns  constructed  on  it,  yet  it  applies  equallj"  to  any 
part  of  the  interval  and  to  the  sums  constructed  on  that  part.  Hence  if  Si  =  -V,-5i  be 
the  part  of  .S  due  to  the  interval  5,-  and  if  S\  be  the  part  of  6"  due  to  this  interval 
after  the  introduction  of  some  of  the  additional  points  into  it,  7H,-5,-  ^  ,S^  s  Si  —  Midi. 
Hence  -S/  is  not  greater  than  .s',-  (and  as  this  is  true  for  each  interval  5,,  .S"  is  not 
greater  than  .S)  and,  moreover,  -S,—  S^  is  not  greater  tlian  0,-5,-  and  a  fortiori  not 
greater  than  OA.  As  there  are  only  ?i'  new  points,  not  more  than  u'  of  the  intervals 
5;  can  be  affected,  and  hence  the  total  decrease  .S  —  S'  in  .S  cannot  be  more  than 
n'OA.    The  treatment  of  .s  is  analogous. 

Inasmuch  as  (^1)  shows  that  the  sums  N  and  s  are  limited,  it  follows  from  Theo- 
rem 4  that  they  possess  the  frontiers  required  in  Theorem  20.  T(^  prove  Theorem  21 
note  first  that  as  L  is  a  frontier  for  all  the  sums  S,  there  is  some  particular  sum  S 
which  differs  from  L  by  as  little  as  desired,  say  I  e.  For  this  .S  let  ji  be  the  number 
of  divisions.  Now  consider  .S'  as  any  sum  for  which  each  5,-  is  less  than  A  =  J  (/nO. 
If  the  sum  .S"  be  cimstructed  by  adding  the  n  points  of  division  for  S  to  the  points 
of  division  for  S'.  ,s"  cannot  l)e  greater  than  S  and  hence  cannot  differ  from  L  by 
so  much  as  I  e.  Also  S"  cannot  be  greater  than  .s"  and  cannot  lie  less  than  .S'  by 
more  than  7(OA.  which  is  \  e.  As  S"  differs  from  L  l>y  less  than  i  e  and  .S"  differs 
from  .s"  by  less  than  ,'  e.  S'  cannot  differ  from  L  Ij}'  nujre  than  e,  which  was  to  be 
pro\ed.    The  treatment  of  .s  and  /  is  analogous. 

29.  If  indices  are  introduced  to  indicate  the  interval  for  Avhich  the 
frontiers  L  and  /  are  calculated  and  if  /8  lies  in  the  interval  from  '/  to  h, 
then  L^f  and  /,f  will  Ix*  functions  of  fi. 

Theorem  22.  The  equations  L^  =  L,1  -f-  Z  *,  n  <r<h;  L^^  =  —  /,,;' ; 
L';  =^  ix{I)  —  (i),  III  ^  fx-^  M,  hold  for  L,  and  similar  ecjiuitions  lor  /.  As 
functions  of  /3.  />,f  and  /,f  are  continuoits,  ami  if  /'(./•)  is  continuous, 
they  are  ditt'erentiable  and  have  the  common  derivative  /{ft)- 

To  prove  tliat  Z,''  =  Z^f  +  />,!'.  consider  c  as  one  of  the  points  of  division  of  the 
interval  from  a  to  '/.  Then  the  sums  -S  will  sati.sfy  .S'^''  =  S^  +  S^,  and  as  the  limit 
of  a  sum  is  the  sum  of  the  limits,  the  corresixjnding  relation  must  hold  for  the 
frontier  L.  To  show  that  L'^  =  —  Z,"  it  is  merely  necessary  to  note  that  Sj*  =  —  .S,^ 
becau.se  in  passing  from  h  to  a,  the  intervals  5;  nmst  be  taken  with  the  sign  opposite 
to  that  which  they  have  when  the  direction  is  from  a  to  h.  From  (^4)  it  appears 
that  m  (h  —  (/)  ^  s'^  ^  M {h  —  a)  and  lience  in  the  limit  in (h  —  n)  ^  Z,''  ^  3/ {h  —  u). 


52  INTRODUCTORY  REVIEW 

Hence  there  is  a  value  fi,  m  ^  /x  ^  M,  such  that  L^^  =  /i{b  —  a).   To  show  that  L^ 
is  a  continuous  function  of  ^,  take  A'  >i^^f  |  and  |nij,  and  consider  the  relations 
Xa  +  A  _  i3  =  i3  +  i3  +  /-  _  LB  ^  LB  +  n  ^  ^^^  |^|<  j,^^ 

LB-k_  LB  ^  Lf  -  L^-"  -  H-n  =- L^-,  ="  m'/^         \f^'\<K. 

Hence  if  e  is  assigned,  a  5  may  be  found,  namely  5  <  e/K,  so  that  |  i,f  ~ ''  —  if  |  <  e 
when  h<d  and  L^f  is  therefore  continuous.    Finally  consider  the  quotients 

L^  +  >'-Lf  L^-''-Lf} 

^  =  11     and 


h  r-  _j^ 

where  ya  is  some  number  between  tlie  maximum  and  minimum  of /(x)  in  the  inter- 
val /3  ^  X  ^  /3  +  /t  and,  if  /  is  continuous,  is  some  value  /(^)  of  /  in  that  interval 
and  where  //  =f{^')  is  some  value  of  /  in  the  interval  ^  —  h^x^  (3.  Now  let 
h  =  0.  As  the  function /is  contiiuious,  lim/(^)  =f{l3)  and  lim/(f')  =f{p).  Hence 
the  right-hand  and  left-hand  derivatives  exist  and  are  equal  and  the  function  L^ 
has  the  derivative /(/3).    The  treatment  of  I  is  analogous. 

Theorem  23.  Eor  a  given  interval  and  function  /,  the  quantities  / 
and  L  satisfy  the  relation  I  ^  L  ;  and  the  necessary  and  sufficient  con- 
dition that  L  =  I  is  that  there  shall  he  some  division  of  the  interval 
which  shall  make  2  (Mi  —  'iii-)  S,-  =  20,8,-  <  e. 

If  L^  =  /^f,  the  function  /'  is  said  to  be  integrable  over  the  interval 

from  a  to  b  and  the  integral    I    /(.<')  d-r  is  defined  as  the  common  value 

X^*  =  J^'^.    Thus  the  definite  integral  is  defined. 

Theorem  24.  If  a  function  is  integrable  over  an  interval,  it  is  inte- 
grable over  any  ])art  of  tlie  interval  and  the  equations 

f(.r)rlr+  j     f(.r)d.r=j    f(.r)dj; 

hold;  moreover,  j  /'(,/•)  r/.r  =  7''(/3)  is  a  continuous  function  of /3 ;  and 
if /'(./•)  is  continuous,  the  derivative  J'''(I3)  will  exist  and  be/(^). 

By  (^-1)  the  sums  ^'  and  ,s  constructed  for  tlie  same  division  of  the  interval  satisfy 
tlie  relation  8  —  .s  ^  0.  By  Uarboux's  Tlieorem  tlie  sums  S  and  s  will  approach  the 
values  L  and  I  when  the  divisions  are  indefinitely  decreased.  Hence  L  —  1^0. 
Now  \i  L  =  I  and  a  A  be  found  so  that  when  5,-  <  A  the  inetjualities  >S'  —  i  <  i  e  aiu.l 
Z  —  .s  <  I  e  hold,  then  N  —  .s  =  S  (.1//  —  m;)  5/  =  20,-5,-  <  e  ;  and  hence  the  condition 
S0,'5,-  <  e  is  seen  to  be  necessary,  ('inversely  if  there  is  any  method  of  division  such 
that  20,-5i  <  e.  tlieii  N  —  .s  <  e  and  the  lesser  (luaiitity  L  —  I  must  also  be  less  than  e. 
But  if  the  difference  between  two  constant  (juautities  can  be  made  less  than  e, 
where  e  is  arbitrarily  assigned,  the  constant  (juautities  are  ecjual  ;  and  hence  the 


FUNDAMENTAL  THEORY  53 

condition  is  seen  to  be  also  sufficient.  To  show  that  if  a  function  is  integrable  over 
an  interval,  it  is  integrable  over  any  part  of  the  interval,  it  is  merely  necessary  to 
show  that  if  i„  =  /„*,  then  L^  =  l^  where  a  and  /3  are  two  points  of  the  interval. 
Here  the  condition  S0,-5,<e  applies;  for  if  S0,-5,-  can  be  made  less  than  e  for  the 
wiiole  interval,  its  value  for  any  part  of  the  interval,  being  less  than  for  the  whole, 
must  be  less  than  e.    The  rest  of  Theorem  2-1  is  a  coi'ollary  of  Theorem  22. 

30.  Theorem  25.  A  function  is  integrable  over  the  interval  n  S  ./■  ^  h 
if  it  is  continuous  in  that  interval. 

Theorem  26.  If  the  interval  a  ^  ./■  ^  h  over  which  f{x')  is  defined 
and  limited  contains  only  a  finite  number  of  points  at  "which  /  is  dis- 
continuous or  if  it  contains  an  iniinite  number  of  points  at  which  /'  is 
discontinuous  but  these  })oints  have  only  a  finite  number  of  points  of 
condensation,  the  function  is  integrable. 

Theorem  27.    If  /'(•'')  ^^  integraljle  over  the  interval  ''  ^  ,/•  ^  h,  the 

sum  or  =  2/'(^,)  8,-  will  approach  the  limit  I  /{•'')  d-''  when  the  indi- 
vidual intervals  8,-  a})proach  the  limit  zero,  it  Ijeing  immaterial  how 
they  apijroach  that  limit  or  how  tlie  points  ^,-  are  selected  in  their 
respective  intervals  S,. 

Theorem  28.    If /"(./•)  is  continuous  in  an  interval  ((^.r^f),  then 

/(.'•)  has  an  indefinite  integral,  namely  |    f(j')dj:,  in  the  interval. 

Theorem  25  may  be  reduced  to  Theorem  23.  For  as  the  function  is  continuous, 
it  is  possible  to  find  a  A  so  small  that  tlie  oscillation  of  the  function  in  any  interval 
of  length  A  shall  be  as  small  as  desired  (Theorem  9).  Suppose  A  be  chosen  so  that 
the  oscillation  is  less  than  e/{l)  —  «).  Then  Z0,-5,-  <  e  when  5,-  <  A  ;  and  the  function 
is  integrable.  To  prove  Theorem  2*!,  take  lirst  the  case  of  a  tiiiite  muuber  of  discon- 
tinuities. Cut  out  the  diseontiiuiities  surrounding  each  value  of  j-  at  wliich/  is  dis- 
continuous by  an  interval  of  length  5.  As  the  oscillation  in  each  of  tiiese  intervals 
is  not  greater  than  0,  the  contribution  of  these  intervals  to  the  sum  ^Oi5i  is  not 
greater  than  On8,  where  n  is  the  mnnber  of  the  discontinuities.  V,y  taking  5  small 
enough  this  may  be  made  as  small  as  desired,  .say  less  than  l  e.  Now  in  each  of  the 
remaining  parts  of  the  interval  a^x^b.  the  function  /  is  contiiuious  and  hence 
integrable,  and  consequently  the  value  of  20,-5i  for  these  portions  may  be  made  as 
small  as  desired,  .say  ^e.  Thus  the  sum  S0,-5;  for  the  whole  interval  can  be  made 
as  .small  as  desired  and/(j")  is  integrable.  When  there  are  points  of  condensation 
they  may  be  treated  just  as  the  isolated  points  of  discontinuity  were  treated.  After 
they  have  been  surrounded  by  intervals,  there  will  remain  over  only  a  finite  num- 
ber of  discontinuities.    Further  details  will  be  left  to  the  reader. 

For  the  proof  of  Theorem  27.  appeal  may  be  taken  to  the  fundamental  relation 
(^1)  which  shows  that  s  ^  a-  ^  S.  Now  let  the  number  of  divisions  increase  indefi- 
nitely and  each  divi.sion  become  indefinitely  small.    As  the  function  is  integrable, 

N  and  ,s  approach  tlie  same  limit  |  f{x)dx.  and  consequently  a  which  is  included 
between  them  must  approach  that  limit.    Theorem  28  is  a  corolla  :-y  (jf  Theorem  24 


54  INTRODUCTOEY  REVIEW 

XX 
f  {x)  dx  in  f  {x) .  By  defi- 
nition, tlie  indefinite  integral  ig  any  function  wliose  derivative  is  tlie  integrand. 
f{x)dx  is  an  indefinite  integral  of /(x),  and  any  other  may  be  obtained 

a 

by  adding  to  this  an  arbitrary  constant  (Theorem  10).  Thus  it  is  seen  that  the 
proof  of  the  existence  of  the  indefinite  integral  for  any  given  continuous  function 
is  made  to  depend  on  the  theory  of  definite  integrals. 

EXERCISES 

1.  Rework  some  of  the  proofs  in  the  text  with  I  replacing  L. 

2.  Show  that  the  L  obtained  from  C'f{x),  where  C  is  a  constant,  is  C  times  theX 
obtained  from/.  Also  if  u,  v,  v:  are  all  limited  in  the  interval  a^x^b,  the  L  for 
the  combination  u  +  v  —  iv  will  be  L  (m)  +  L{v)  —  L  (w),  where  L  (u)  denotes  the  L 
for  u,  etc.  State  and  prove  the  corresi)onding  theorems  for  definite  integrals  and 
hence  the  corresponding  theorems  for  indefinite  integrals. 

3.  Show  that  SOjSj  can  be  made  less  than  an  assigned  e  m  the  case  of  the  func- 
tion of  Ex.  6,  p.  44.  Note  that  /  =  0,  and  hence  infer  that  the  function  is  integrable 
and  the  integral  is  zero.  The  proof  may  be  made  to  depend  on  the  fact  that  there 
are  only  a  finite  muuber  of  values  of  the  function  greater  than  any  assigned  value. 

4.  State  with  care  and  prove  the  results  of  Exs.  3  and  5,  p.  20.  "What  restric- 
tion is  to  be  placed  onf{x)  if /(^)  may  replace  /x  ? 

5.  State  with  care  and  prove  the  results  of  Ex.  4,  p.  29,  and  Ex.  13,  p.  30. 

6.  If  a  function  is  limited  in  the  interval  a^x^h  and  never  decreases,  show 
that  the  function  is  integrable.    This  follows  from  the  fact  that  20,-  s  0  is  finite. 

7.  More  generally,  \etf{x)  be  such  a  function  that  ZO,-  remains  less  than  some 
number  A',  no  matter  how  the  interval  be  divided.  Show  that/  is  integrable.  Such 
a  function  is  called  a  function  of  limited  variation  (§  127). 

8.  (Jhange  of  variable.  Let  f{x)  be  continuous  over  a^x^  b.  Change  the 
varialjle  to  x  =  4>{t).  where  it  is  supjjosed  that  a  =  4>{t^)  and  b  =  0(io),  and  that 
<p(t),  4>'{t),  s.n(\f[(p{t)]  are  continuous  in  t  over  t^^t^  t„.    Show  that 

Cf{x)dx^  f'y[<p{t)]<p'{t)dt     or      f'^^'^f{x)dx=f[n4>{t)]4'\t)dt. 

Do  this  by  showing  that  the  derivatives  of  the  two  sides  of  the  last  equation  with 
respect  to  t  exist  and  are  e(iual  over  t^  ^t^t.,,  that  the  two  si(h's  vanish  when 
t  =  i^  and  are  equal,  and  hence  that  they  nmst  be  eiiual  tliroughout  the  interval. 

9.  Osgood'^  Theorem.  Let  n-;  be  a  set  of  (luantities  which  differ  uiiifonuly  from 
f{ki)  St  by  an  amount  f,-5i,  that  is,  suppose 

ai  =f{^;)  di  +  ^i5i,     where     \^i\<e     and     «  ^  ?  ^  b. 
Trove  that  if /is  integrable,  the  sum  2«j  approaches  a  limit  when  5;  :^  0  and  that 
the  limit  of  the  sum  is    I    f{x)dx. 

.     10.   Ai)i)lv  Ex.  9  to  the  case  Af  =  f'Ax  +  ^Ax  where/'  is  continuous  to  show 

r'' 
directly  that /(/*)  -/(//)  =  I    f'{x)dx.   Also  by  regarding  Aj:  =  <p'  {()  At  +  fA/.  apply 

to  Ex.  8  to  xjrove  the  rule  for  change  of  variable. 


PART  I.    DIFFERENTIAL  CALCULUS 

CHAPTER   III 

TAYLOR'S   FORMULA  AND  ALLIED   TOPICS 

31.  Taylor's  Formula.  The  obje(;t  of  Taylor's  Formula  is  to  express 
the  value  of  a  function  f(j')  in  terms  of  the  values  of  the  function  and 
its  derivatives  at  some  one  point  x  =  <i.    Thus 

/(•^•)  =/(^0  +  C'^  -  ^0./"(")  +  ^^f^/"(^0  +  •  •  • 

Such  an  expansion  is  necessaril\'  true  because  the  remainder  R  may  be 
considered  as  defined  by  the  equation ;  the  real  significance  of  the 
formula  must  therefore  lie  in  the  ])ossibility  of  finding  a  simple  ex- 
pression for  R,  and  there  are  several. 

Thkokkm.  On  tlie  hypothesis  that  /{■>•)  and  its  first  n  derivatives 
exist  and  are  continuous  over  the  intei'val  a  ^  .'•  ^  b,  the  function  may 
be  expanded  in  that  interval  into  a  })olynomial  in  x  —  a, 

/(■'■)  =,/■(")  +  (■'■  -  ")/"(,")  +  ^-^^^^T^f'X")  +  ■■■ 

with  the  remainder  7?  expressible  in  any  one  of  the  forms 

(x  —  ay  J/"  (1  —  0)"-'^ 

where  h  =  x  —  a  and  a  <  t  <  x  or  ^=  <i  ■\-  Bh  where  0  <  ^  <  1. 

A  first  proof  may  be  made  to  depend  on  Kolle's  Theorem  as  indicated  in  Kx.  8, 
p.  4!).    Let  X  be  regarded  for  the  moment  as  constant,  say  equal  to  b.    Constrnct 

55 


56  DIFFERENTIAL  CALCULUS 

the  function  \p  {j)  there  indicated.  Note  that  \}/  («)  =  \l/  (h)  =  0  and  that  the  deriva- 
tive ^'(x)  is  merely 

n^)=-~ ^/^"H-^)  +  n\ ^~-\  f{h)-f{a)  -  {h  -  a)f'{a) 

(n-l)I  (6-  «)«    L 

(H  -  1)  !  '  ^  ^J 

By  llolle's  Tlieoreni  \p'{^)  =  0.    Hence  if  ^  be  substituted  above,  the  result  is 

f{h)  =f{a)  +  {h  -  ,,)/'(")  +  •  •  •  +  ^"^J"^V("-^>('0  +  ^'^^V<">(f), 

(7t  —  1)  !  n\ 

after  strikini;'  out  the  factor  —  {!)  —  f)"  -i,  nudtiplyin^-  l)y  (/;  —  aY/n^  and  transposing 

/(/;).    The  tlieoreni  is  therefore  proved  with  the  first  form  of  the  remainder.    This 

proof  does  not  require  the  continuity  of  the  nth  derivative  nor  its  existence  at  a  and  at  b. 

The  second  form  of  the  remainder  may  be  found  by  api)lyiug  llolle's  Theorem  to 

^  {X)  =f{b)  -  /(./•)  -  (/>  -  x)r{x) -'r^Tvr  •^■'" "' '  (■'■)  -  (''  - '")  ^' 

(u-l)  ! 

where  P  is  deternuned  so  that  li  =  {!>  —  a)  P.  Note  tliat  i/- (/;)  —  0  and  that  by 
Taylor's  rormula  i/'  (a)  =  0.    Now 

^V)---:  ----/<"H.'-)  +  ^'   <>!•   ^=/^"H0S— "rrr   ^'"^■''   ^'(^')  =  o. 

(H-f)!  ("-!)! 

Hence  if  |  be  written  ^  =  a+0h  where  h  =  l>—a,  then  J>—^  =  h—  a—  6h.=  {J>—  a){\  —  d) . 

And      R  =  {b-  a)  P  =  {b-a)  ^ 1--  ^^^^-fOO^  =  i /\,  f""  i^)- 

{n  —  1)  !  (ji  —  1)  ! 

The  second  form  of  R  is  thus  found.  In  this  work  as  before,  the  result  is  proved 
for  X  =  b,  the  end  point  of  the  interval  a^x^t).  But  as  tlu^  interval  could  be 
considered  as  terminating  at  any  of  its  points,  the  proof  clearly  applies  U>  any  x 
in  the  interval. 

A  second  jiroof  of  Taylor's  Formula,  and  the  easiest  to  remember,  consists  in 
integrating  the  jtth  derivative  n  times  from  a,  to  x.    The  successive  results  are 

f  "/-(.o  (,,■)  dx  =  /»  - 1  (X)  ]"=  /(«  - 1 )  (X)  -  /("  - 1)  (a) . 

f  r7<")(x)(/./;2  =  r7(«-i)(j:) (/,,._  c'foi-i)(^,t)dx 


f-  •  •    r>'0(x)(Zx«  =/(x,  -/(.A)  -  (X  -  a)r{a) 

I'll  'J  a 


2  !  (n  —  1)  ! 


'I'lie  formula,  is  therefore  jjroved  with  R  in  the  form    |     •  •  •  /     /(")(x)(?x".   To  trans- 
form this  t,o  the  ordinary  form,  the  Law  of  t,he  Mean  may  l)e  applit'd  ((<>•">),  S  1<»)-    ^'''"' 

m(x-«)<  rV<")(x)tZx<.V(x~«),       w^^-.ZL.-')!<  T'...  f /X'0(x)(Zx'' <3/^--^-"-^-, 

^«  ?i  !  t/«  J  a  V.  ! 


TAYLOR'S  FORMULA;  ALLIED  TOPICS  5T 

where  m  is  the  least  and  M  the  greatest  vahie  of /(")(j)  from  a  to  x.   There  is  tlien 
some  intermediate  valne/(")(^)  =  ^  such  tliat 

J  a  J  a  n\ 

Tliis  prooi'  requires  that  tlie  nth  derivative  be  continuous  and  is  less  general. 
The  third  proof  is  obtained  by  applying  successive  integrations  by  parts  to  the 

Jr\  h 
/'{a  +  h  —  t)  dt  to  make  the  integrand  contain 
u 
liigher  derivatives. 

f{a  +  Ji)  -  /(«.)  =  f  \f'{a  +  A  -  0  dt  =  tf'{a  +  h  -  t)\''  +  f ''  if '{a  +  h  -  t)  dt 

Ju  Jo         «^0 

=  hfia)  +  I  t'f'\<^  +  li-t)X  +  f''  h  i:y""{a  +  //.  -  t)  dt 

=  hfia)  +  ^^r{a)  + .  .  .  +  T^^,/("  ~'\n)  +  (  "-^^ '~--i/(")(a  +  h-t)  dt. 
2  !  (u  —  1)1  i/o    (h—  1)1 

This,  however,  is  precisely  Taylor's  Fornmla  with  tlie  third  form  of  remainder. 

If  the  point  a  about  which  the  function  is  expanded  is  .r  =  0,  tlie 
expansion  Avill  take  the  fonu  known  as  JMaclauriirs  Formula: 

/(•O  =/(0)  +  ^■■fi^  +  ;;';/"(0)  +■ . .  +  j£^.r"'^''  (0)  +  n,  (3) 
P'^X6.r)  =  -^  (1  -  ey  ~\f<"\e.r)  =  j--^jyf  '^"  -\f"%^  -  0  '^^- 

32.  Both  Taylor's  Formula  and  its  special  cas(>,  ]\raclaiirin*s,  express 
a  function  as  a  polynomial  in  //  =  .'■  —  r',  of  which  all  the  coefhcients 
except  the  last  are  constants  while  the  last  is  not  constant  but  depends 
on  //  both  explicitly  and  throiigh  the  unknown  fraction  Q  which  itself  is 
a  function  of  h.  If,  however,  the  nxX\  derivative  is  continuous,  the  coeffi- 
cient/'^"^(^^  +  ^//)//i !  must  remain  finite,  and  if  the  form  of  the  deriva- 
tive is  known,  it  may  be  possible  actually  to  assig'n  limits  between 
Avhich /'"^"^(rf  +  6//)/«  !  lies.  This  is  of  gri'at  importance  in  making- 
approximate  calculations  as  in  Exs.  8  if.  below ;  for  it  si'ts  a,  limit  to 
tlie  value  of  R  for  any  value  of  n. 

TiiKoi;K:\r.  There  is  only  one  possible  expansion  of  a  function  into 
a  polynomial  in  //  =  ./•  —  d  of  which  all  the  coefficients  except  the  last 
are  constant  aiul  the  last  finite  ;  and  lu^nce  if  such  an  expansion  is 
found  in  any  mariner,  it  must  be  Taylor's  (or  jMaclaurin's). 

To  prove  this  theorem  consider  two  polynomials  of  the  nth  order 
Co  +  ''\j^  +  ^-]>^^  + 1-  t'n-i/i""^  +  t^H/t"  =  C'y  -i-  6\/t  +  CJiP-  + 1-  C'„_i//,«-i  +  ('„/<", 

which  represent  the  same  function  and  hence  are  eipial  for  all  values  of  /t  from  0 
to  6  —  «.    It  follows  that  the  coefficients  nmst  be  equal.    For  let  h  approacli  0. 


'■■  =  ;;: 


58  DIFFER EXTIAL  CALCULUS 

The  terms  containing  h  will  ai^proach  0  and  hence  Cq  and  Cq  may  be  made  as 
nearly  eqnal  as  desired  ;  and  as  they  are  ct)nstants,  they  must  be  equal.  Strike 
them  out  from  the  equation  and  divide  hj  h.  The  new  equation  must  hold  for  all 
values  of  h  from  0  to  b  —  a  with  the  possible  exception  of  0.  Again  let  h  =  0  and 
now  it  follows  that  c^  —  G'j.  And  so  on,  with  all  the  coefficients.  The  two  devel- 
opments are  seen  to  be  identical,  and  hence  identical  with  Taylors. 

To  illustrate  the  applicatit)n  of  the  theorem,  let  it  be  required  to  find  the  expan- 
sion of  tanx  about  0  when  the  expansions  of  sinx  and  cosx  about  0  are  given. 

sin  X  =  X  —  ^  x^  +  jl  ^j  x^'  +  Px',  cos  X  =  1  —  i  X-  -f-  ^'j  x'*  +  Qx*^, 

where  P  and  Q  remain  finite  in  the  neighboi'hood  of  x  =  0.  In  the  first  place  note 
that  tanx  clearly  has  an  expansion  ;  for  the  function  and  its  derivatives  (which 
are  combinations  of  tan  x  and  sec  x)  are  finite  and  contiiuious  until  x  approaches  i  tt. 
By  division, 

X    -I-    1  X^    +      y---     X^ 

1  -  I  X2  +  2\  X*  +   QX«)  X  -   1  X'^  +  t' 0  ^'  \  +  ^^'' 

X  -  J-  x»  +  IT  T  -c^ ;  +  Q^'' 


,3-3_    1    X":  +  -Sx'  +  1QX« 


Hence  tan  x  =  x  4-  J  a""  +  tVx''  -I x',  where  S  is  the  remainder  in  the  division 

cosx 
and  is  an  expression  containing  P,  Q,  and  powers  of  x  ;  it  must  remain  finite  if  P 
and  Q  remain  finite.    The  quotient  .S'/cos  x  which  is  the  coefficient  of  x'  therefore 
remains  finite  near  x  =  0,  and  the  expression  for  tan  x  is  the  Maclaurin  expansion 
u})  to  terms  of  the  sixth  order,  plus  a  remainder. 

In  the  case  of  functions  compounded  from  simple  functions  of  which  the  expan- 
sion is  known,  this  method  of  obtaining  the  expansion  by  algebraic  processes  upon 
the  known  expansions  treated  as  polynomials  is  generally  shorter  than  to  obtain 
the  result  by  differentiation.  The  computation  may  be  abridged  by  omitting  the 
last  terms  and  work  such  as  follows  the  dotted  line  in  the  example  above  ;  but  if 
this  is  done,  care  nuist  be  exercised  against  carrying  the  algebraic  operations  too 
far  or  not  far  enough.  In  Ex.  .5  below,  the  last  terms  should  be  put  in  and  carried 
far  enough  to  insure  that  the  desired  expansion  has  neither  more  nor  fewer  terms 
than  the  circumstances  warrant. 

EXERCISES 

1.  Assume  E  =  (b  -  a)>^P:  show  R  =  —  ^^  ~  ^^''~  V^"'' (^^)- 

(n-l)'.lc 

2.  Apply  Kx.  o,  p.  2i>,  to  compare  the  third  form  of  remainder  with  the  first. 

3.  Obtain,  by  differentiation  and  suljstitution  in  (1),  three  nonvanisliing  terms: 

(a)  sin-'x.  a  =  0.        (/i)  tanh  x,  a  =  0,  (7)  tanx.  «  =  1  tt, 

(5)  cscx,  a  —-  }.  TT.        (e)  e*'"'',  a  =  0,  {^)  log  sin  x.  it  =  Iw. 

4.  Find  the  )(th  derivatives  in  the  following  cases  and  write  the  expansion: 

(((-)  sin  X.  ((  —  0,  ip)  sin  x.  ((  =  I  tt,  (7)  r'\  d  =  0. 

(5)  c',  a  =  1,  (e)  logx,  a  ^  1,  (f)   (1  +  £)''\  "■  -  0. 


TAYLOR'S  FORMULA;  ALLIED  TOPICS  59 

By  algebraic  processes  find  the  Maclaurin  expansion  to  the  term  in  x^ : 


(a)  sec  x,  (^)  tanh  x,  (7)   —  Vl  —  x-, 

(5)  e^siiix,  (e)  [log(l  — x)]-,  (f)   +  Vcosh  x, 

(17)  e«i°^,  (ff)  hjgcosx,  (t)  log  Vl  4- X-. 

The  expansions  needed  in  this  worlv  may  be  foinid  by  differentiation  or  taken 
from  B.  O.  Peirce's  "Tables.""  In  (7)  and  (f)  apply  the  binomial  theorem  of  V.x. 
4  (f).  In  {r])  let  y  =  sinx,  expand  e'J,  and  substitute  for  y  the  expansion  of  sinx. 
In  (d)  let  cosx  =  1  —  y.  In  all  cases  show  that  the  coefficient  of  the  term  in  x^ 
really  remains  finite  when  x  =  0. 

6.  If  f{a  +  h)  =  Cy  +  c-j/i  +  c.Ji-  +  ••  •  +  (•„_!/(" -1  +  c„/(",  show  that  in 

f  f{a  +  h)  dh  =  cji  +  -1  h"  +  ^  AJ5  +  .  , .  +  'hl^  kn  _^  C  'r,JiHlh 
Jo  2  3  n  Jo 

the  last  term  may  really  be  put  in  the  form  Pit"  +1  with  P  finite.    Apply  Ex.  5,  p.  29. 

_  r  ^      (If 

7.  Applv  Ex.  0  to  sin-ix  =  j     ,   etc..  to  find  developments  of 

Jo    Vl  -  X- 

(a)  sin  -1 X,                     (/3)  tan-i  x,                     (7)  sinh-i  x, 
(5)  log:^,  (e)     /     C--V/X,  (n     /      dx. 

1  —  X  J()  J<}  X 

In  all  these  ca.ses  the  results  may  be  found  if  desired  to  n  terms. 

8.  Show  that  the  remainder  in  the  Maclaurin  development  of  e^  is  less  than 
x"e^/n  1 ;  and  hence  that  the  error  introduced  by  disregarding  the  remainder  in  com- 
puting e-'"  is  le.ss  than  x"e-''/n  I.  How  many  terms  will  suffice  to  compute  e  to  four 
decimals'?    How  many  for  c^'  and  for  e"-^  ? 

9.  Show  that  the  error  introduced  by  disregarding  the  remainder  in  comput- 
ing log  (1  +  x)  is  not  greater  than  x"/n  if  x  >  0.  Ibiw  mauy  terms  are  required  for 
the  computation  of  log  1^  to  four  places?    of  log  1.2  '.'    Compute  tlie  latter. 

10.  The  hypotemise  of  a  triangle  is  20  and  one  angle  is  ol".  Find  tlie  sides  by 
expanding  sinx  and  cosx  about  a  =  i  ir  as  linear  functions  of  x  —  ^  tt.  Examine 
the  term  in  (x  —  l-rr)-  to  find  a  maximum  value  to  the  error  introduced  by 
neglecting  it. 

11.  Compute  to  6  places:  (a)  ti  (/3)  log  1.1.  (7)  sin  .30'.  (5)  cos  30'.  During 
the  computation  one  place  more  than  the  desired  number  should  be  carried  along 
in  the  arithmetic  work  for  safety. 

12.  Show  that  the  remainder  for  log  (1  +  x)  is  less  than  x"/)i  (1  +  x)"  if  x  <  0. 
Compute  (a)  log  0.9  to  0  places,   (^)  log  0.8  to  4  places. 

13.  Show  that  the  remainder  for  tan-i/  is  less  than  x"/n  where  ?i  may  always 
be  taken  as  odd.    Compute  to  4  places  tan-i  i. 

14.  The  relation  |  tt  =  tan-i  1  =  4  tan-'  I  —  tan-i  ^ It  enables  1  tt  to  be  found 
easily  from  the  series  for  tan-^x.  Find  ^  tt  to  7  places  (intermediate  work  carried 
to  8  places). 

15.  Computation  of  logarithms,    (a)  If  a  =  log  1,,".  /*  =  log  |i,  c  =  log  Ji.  then 
loff  2  =  7  (/  -  2  6  -1-  3 c,         lo£t  3  =  11  «  -  3  h  +  5 c.         \> >g  5  =  10  a  —  ib  +  7  c. 


60 


DIFFERENTIAL  CALCULUS 


Now  a  =  —  log  (1  —  j\,),  b  =—  log  (1  —  y^o),  c  =  log  (1  +  -jjig)  are  readily  computed 
and  hence  log  2,  log  3,  logo  may  be  found.  Carry  the  calculations  of  a,  5,  c  to 
10  places  and  deduce  the  logarithms  of  2,  3,  5,  10,  retaining  only  8  places.  Com- 
pare Puirce's  "Tables,"  p.  109. 

1  +  X  2  .r" 

(S)  Show  that  the  error  in  tlic  series  for  loij; is  less  than '- Com- 

^  '  '    1  -  X  )i  (1  -  x)« 

pute  log  2  corresponding  to  x  =  -^  to  4  places,  log  1|  U)  5  places,  log  l'^  to  0  places. 


(y)  Show  log 


Ip  +  <i 


3  \?)  +  ql 


+ 


+  llm 


<1  LP  +  'I      ii  \P  +  q/  2  H  —  1  \p  +  q/ 

give  an  estimate   of  li-2>i^i-  <^'i'l  compute  to  10  figures  log  3  and  log  7  from  log  2 
and  loi;-  5  of  Tcirce's  "Tables  "  and  from 


4  log  3—4  lo','-  2  —  If  )Lr  5  =  loi 


81 


41( 


r,  =  loir- 


Ii„< 


80  '-'  '7^-1 

16.  Compute  Ex.  7  (e)  to  4  places  for  x  =  1  and  to  G  places  for  x  =  |. 

17.  Compute  .'in-iO.l  to  seconds  and  sin-i  ?,  to  minutes. 

18.  Show  that  in  tlie  expansion  of  (1  +  x)^'  tlie  remainder,  as  x  is  >  or  <0,  is 

k-{k-l)---{k-n  +  ])         X' 


\k.{k-l)...{k-n  +  l)  ^„^ 
1  •  2...  ?i 


or   /?„< 


',  n>k. 


1-2..-71  (l  +  x)«- 

Hence  compute  to  5  figures  \^10;J,  V!i8,  ^28,  V2.j0,  a'ioIJO. 

19.  Sometiiues  the  remainder  cannot  be  readily  found  but  tl;e  terms  of  the 
expansion  appear  to  be  dinnnisliing  so  rapidly  that  all  after  a  t-ertain  point  appear 
negligible.  Thus  use  Peirce"s  "Tables,"  Nos.  774-789,  to  comi>ute  to  four  places 
(estimated)  tlie  values  of  tan  0^  log  cos  10',  esc  3^^,  sec  2'^. 

20.  Find  to  witliiu  1'^,'.  the  ari'a  under  cos  (x-)  and  sin  (x-)  from  0  to  ^  tt. 

21.  ^V  unit  magnetic  polo  is  j.l^H'cd  at  a  distance  L  from  the  center  of  a  magnet 
of  pole  strengtli  J/  and  Icngt'.i  2/.  wliere  l/L  is  small,  i'ind  the  force  on  the  ])ole 
if  (a)  the  pole  is  in  the  line  of  the  magnet  and  if  {^)  it  is  in  the  perpendicular 
bisector. 


Ans.   (a)   -^^  (1  +  e)  with  e  about  2( 


{3)   "  V  (1  —  f)  with  e  ab.iut  '-'  (  - 


22.  The  fornnila  for  the  di.-tance  of  the  horizon  is  J) —\'^h  wlicre  T)  is  the 
distance  in  nnles  and  h.  is  tlie  altitude  of  the  observer  in  feet.  I'rove  the  fonnula 
and  show  that  the  error  is  aliout  l\',  for  heights  up  to  a  few  mih  s.  Take  tlie  radius 
of  the  earth  as  3900  miles. 

23.  Find  an  aiiprnximate  foi-nnda  for  tlio  dip  of  the  liorizou  in  minutes  below 
the  iKU'izontal  if  11  in  feet  is  the  height  of  the  observer. 

24.  If  N  is  a  eircr.lai'  arc  and  ('  its  chord  and  r  tlu;  chord  of  half  the  arc,  prove 
,S'  =  I  (8r  -  (■)  (1  +  f)  where  f  is  aliout  syii\S{)  /.'*  if  /.'  is  the  radius. 

25.  If  tv.'o  ijiiantities  differ  from  each  other  by  a  small  fraction  e  of  their  value, 
show  that  their  geometric  mean  \\ill  differ  from  their  arithmetic  mean  by  about 
\  i-  of  its  value. 

26.  The  algebraic  method  may  be  applied  to  finding  expansioiis  of  some  func- 
tions which  become  infinite.  ('I'lius  if  the  series  fur  cos  x  and  >i:ix  be  divided  to 
find  cot  X,  the  initial  term  is  1/x  and  becomes  infinite  at  x  =  0  just  as  cotx  does. 


TAYLOR^S  FOKMULA;  ALLIED  TOPICS  61 

Such  expansions  are  not  Maclaurin  developments  Init  are  analogous  to  them. 
The  function  xcotx  would,  however,  have  a  ^laclauriu  development  and  tlie 
expansion  found  for  cot  x  is  this  development  divided  by  x.)  Find  the  develop- 
ments about  J  =  0  to  terms  in  x*  for 

(a)  cot  J,  (i3)  cot-j-,  (7)  CSC  J,  (5)  csc^j:, 

(e)  cot  X  CSC  J,  (f)  l/(tau-ij)-,  (7;)   (sinx  —  taux)-i 

27.  Obtain  the  expansions: 
(a)  logsinx  =  logx  — ^x-  — ji^x^  + /i,       (^)  log  taux  =  logx  +  ix- +  g'gX*  +  •  •  ■, 
(7)  likewise  for  log  vers  x. 

33.  Indeterminate  forms,  infinitesimals,  infinites.  If  two  functions 
/'(./')  and  4>{.i-)  are  defined  for  ./■  =  <i  and  if  <^(''')  ^  0,  the  quotient /'/<^  is 
defined  for  ,r  =  a.  I>ut  if  <^  ('")  =  0,  the  quotient  /'/</>  is  not  defined  for  a. 
If  in  tliis  case/'  and  <^  are  defined  and  continuous  in  the  neif,dd)orhood 
of  a  and/(V/)  4^  0,  the  quotient  Avill  become  infinite  as  ,/•  =  a  ;  whereas 
if /(r/)  =  0,  the  behavior  of  the  quotient /'/(^  is  not  immediately  appar- 
ent but  gives  rise  to  the  indeterminate  form  0/0.  In  like  manner  if  / 
and  <^  liecome  infinite  at  a,  the  quotient  f  /<^  is  not  defined,  as  neither 
its  numerator  nor  its  denominator  is  deiined ;  thus  arises  the  indeter- 
minate fornt  x/x.  The  question  of  determining  or  evaluating  an 
indeterminate  form  is  merely  the  (piestion  of  finding  out  whether  the 
(piotient  /'/</>  approaches  a  limit  (and  if  so,  what  limit)  or  l)eco]ues 
positively  or  negatively  infinite  when  ./•  approaches  ". 

Tii?:oki;m.  L' lI(i.<p\f(iTs  Riih'.  If  the  functions  /'(./•)  and  <^  (.'•),  which 
give  rise  to  the  indeterminate  form  0/0  or  x/x  Avlien  ./•  =  '/,  arc;  con- 
tinuous and  differentiable  in  the  interval  (t  <  .'■  =s  ^>  and  if  //  can  l)e 
taken  so  near  to  <i  that  <^'('')  does  not  vanish  in  the  interval  and  if  the 
quotient/''/^'  of  the  derivatives  ap])roaches  a  limit  or  l)ecomes  posi- 
tively or  negatively  infinite  as  ./•  =  ",  then  the  quotient  /'/<^  will  ap- 
proach that  limit  or  l)ecome  positively  or  negatively  infinite  as  the  ease 
may  be.  Hence  (ni  indi'fcrmbv'fi;  fnnii  0/0  or  x/x  imn/  he  fcplaci'd  Jnj 
the  (iitiiflinit  of  the  drrlrdfu'es  af  nmni'i'dfoi'  (ni<l  (Jcnnm inntnr. 

C-Vsi;  I.  f{a)  =  (p  {(i)  =  0.   The  proof  follows  from  Cauchy's  Fornuila,  Ex.  (i,  p.  49. 

i (»r = = ,  (I  <  ^  <X. 

<P  (X)        4>  (X)  -  <p  (r/)         0'(?) 

Now  if  X  =  (I.  so  nuist  J.  which  lies  between  x  and  a.  Hence  if  the  quotient  on  tlie 
right  approaches  a  limit  or  becomes  positivelj^  or  negatively  infinite,  the  same  is 
true  of  that  on  the  left.  The  necessity  of  inserting  the  restrictions  that  /  and  0 
shall  be  continuous  and  differentiable  and  that  </>'  shall  not  have  a  root  indefinitely 
near  to  a  is  apparent  from  the  fact  that  Cauchy's  Formula  is  proved  only  fer  func- 
tions that  satisfy  these  conditions.  If  the  ilerived  form/'/^'  should  also  be  inde- 
terminate, the  rule  could  again  be  ajiplied  and  tlie  ^[wnx'u'wi  f" /cf>"  would  replace 
f  /<p'  with  the  understanding  that  pn.iper  restrictions  were  satisfied  by/',  0',  aiul  0". 


62  DIFFEKEXTIAL  CALCULUS 

Cask  II.  f{a)  =  <p{a)  =  x.    Apply  Cauchy's  Formula  as  follows : 

f{x)-f{h)  ^f{.c)    1 -/(/>) //(x)  ^n^)^  a<x<b, 

<t>{z)-<t>{b)      <p{x)  l-.p{b)/<p{x)      4>'{^)'  X  <!</>, 

where  the  middle  expression  is  merely  a  different  way  of  writinij  the  first.  Now 
suppose  that/'(x)/0'(x)  approaches  a  limit  when  .c  =  a.  It  nmst  then  be  possible  to 
take  h  so  near  to  a  t\mtf'{^)/<p'{^)  differs  from  that  limit  by  as  little  as  desired,  no 
matter  wliat  value  ^  may  have  between  a  and  b.  Now  as  /  and  (p  become  infinite 
when  X  =  a,  it  is  possible  to  take  x  so  near  to  a  that  f{b)/f(x)  and  (p  {b)/<p  {x)  are 
as  near  zero  as  desired.  The  second  equation  above  then  shows  that /(x)/0(x), 
multiplied  by  a  quantity  which  differs  from  1  by  as  little  as  desired,  is  etjual  to 
a  quantity  f'{^)/<p'{^)  which  differs  from  the  limit  of  f'{x)/(p'{x)  as  x  =  a  by  as  little 
as  desired.  Hence //^  must  approach  the  same  limit  as/V0'.  Similar  reasoning 
would  apply  to  the  supposition  that/y0'  became  positively  or  negatively  infinite, 
and  the  theorem  is  proved.  It  may  be  noted  that,  by  Theorem  10  of  §  27,  the  form 
/'/<!>'  is  sure  to  be  indeterminate.  The  advantage  of  being  able  to  differentiate 
therefore  lies  wholly  in  the  possibility  that  the  new  form  be  more  amenable  to 
algebraic  transformation  than  the  old. 

The  other  indeterminate  forms  0-  x,  0\  l"".  xf,  ao  —  oo  may  be  reduced  to  the 
foregoing  by  various  devices  which  may  be  indicated  as  follows  : 

0-x  =  -:=-,     Oo^eiogoo^^goiogo  =  eO--.     •••,     oc- oo  =  loo-e- -«>  =  loi;  — . 

^    2_  .       .  ^  .- ^^ 

00  0 

The  case  where  the  variable  becomes  infinite  instead  of  approaching  a  finite  value 
a  is  covered  in  Ex.  1  below.  The  theory  is  tlierefore  completed. 

Two  methods  which  frequently  may  be  used  to  shorten  the  work  of  evaluating 
an  indeterminate  form  are  the  method  of  E -functions  and  tJic  application  of  Taylor's 
Fonnula.  By  definition  an  E-f unction  for  the  point  x  =.  a  is  any  continuous  function 
which  apjjroaches  a  finite  limit  other  than  0  u:hen  x  =  a.  Suppose  then  that/(x)  or 
(p{x)  or  lioth  may  be  written  as  the  products  £",/,  and  E.-,4>^.  Then  the  method  of 
treating  indeterminate  forms  need  be  applied  only  to/j/^^  and  the  result  multiplied 
l)y  lini  Ej/E.,.    For  example, 

X  —  ff  X  o, 

lini  (x-  +  ax  +  a")  lim  — —  =  .3  a-  lim —  =  .3  a-. 

.r  =  aSin(x  —  «)         .7~a  ,,■  =  r,  slu  (x  —  r()  ^.  j,  „  sill  (X  —  c) 

Again, suppose  that  in  the  form  0/0  Imth  numerator  and  denominator  may  be  de- 
veloped about  X  =  a  by  Taylor's  Formula.    The  evaluation  is  immediate.    Thus 

tanx  —  sinx  _  (x  +  ix"  +  Px")  -  (x  —  ^  x^  +  Qx-')  _  I  +  (P  —  Q)x- 
x"  log  (1  +  X)  ^  X-  {x-  i  X-  +  /i'x-5)  ~  1  _  1  X  +  lix-  ' 

and  now  if  x  =  0,  the  limit  is  at  once  shown  to  be  simply  I. 

When  the  functions  l)ecoiiie  infinite  atx  =  a.  the  conditions  requisite  for  Taylor's 
Formula  are  not  present  and  there  is  no  Taylor  expansion.  Nevertheless  an  expan- 
sion may  sometimes  be  obtained  by  the  algebraic  method  ($  32)  and  may  fre(iuentl.v 
be  used  to  advantage.  To  illustrate,  let  it  be  required  to  evaluate  cot  x  —  1/x  which 
is  of  the  form  x  —  x  when  x  =  0.    Here 

cosx       1  +  1,  X-  4-  /V^       1  1  -  1.  ,/•-  4-  /V-"       ]  /         1    „ 

cot  X  = = = -:i =  -    1 X-  +  'Sx' 

sin  X      X  —  I  X-'  +  (^x-J      X  1  —  J  X-  +  (^'x'*      X  \        3 


TAYLOirS  FORMULA;  ALLIE])  TOPICS  63 

where  S  remains  finite  when  x  =  0.    If  this  vahie  be  substituted  for  cot  x,  then 
lini  (cotx )  =  lim  ( x  +  Sx^ )  =  iini  ( x  +  SxA  =  0. 

x  =  o\  Xf         x  =  o\x        3  Xf         x~(j\       3  / 

34.  An  infinitesimal  is  a  variable  vjJiicli  is  ultiwatcl ij  in  aiqtroacli  the 
Hill  it  zero  :  an  infinite  is  a  variable  w/iirh  is  to  become  eitlter  positively 
or  negatively  infiyiite.  Thus  the  increments  A//  and  A.r  are  finite  quan- 
tities, but  when  they  are  to  serve  in  the  definition  of  a  derivative  the}- 
niust  ultimately  approach  zero  and  hence  may  be  called  infinitesimals. 
The  form  0/0  represents  the  quotient  of  two  infinitesimals  :  *  the  form 
x/cc,  the  quotient  of  two  infinites;  and  0-  x,  the  pi'oduct  of  an  infin- 
itesimal by  an  infinite.  If  any  infinitesimal  a  is  chosen  as  the  j>ri ma ry 
infinitesimal,  a  second  infinitesimal  (i  is  said  to  be  of  the  same  order  as 
a  if  the  limit  of  the  quotient  ^/a  exists  and  is  not  zero  wlien  «  =  0 ; 
whereas  if  the  quotient  ^/a  becomes  zero,  /?  is  said  to  be  an  infinites- 
imal of  higher  ordi'r  than  a,  but  of  loirrr  order  if  the  quotient  becomes 
infinite.  If  in  particular  the  limit  /8/V'  exists  and  is  not  zero  when 
a  ==  0,  then  ^  is  said  to  be  of  the  nth  order  relatire  to  a.  The  deter- 
mination of  the  order  of  one  infinitesimal  relative  to  another  is  there- 
fore essentiall}-  a  problem  in  indeterminate  forms.  tSimilar  definitions 
may  be  given  in  regard  to  infinites. 

Theokem.  If  the  quotient  ^/a  of  two  infinitesimals  approaches  a 
limit  or  becomes  infinite  when  «  =  0,  the  qtiotient  /S'/a'  of  two  infin- 
itesimals which  differ  respectively  from  (i  and  a  \)\  infinitesimals  of 
higher  order  will  approach  the  same  limit  or  l)ecome  infinite. 

Theokem.  DahameVs  Theorem.  If  the  sum  Sa,  =  t^ -f  ct, -|- •  •  •  +  «t„ 
of  n  positive  infinitesimals  a]>iiroa('hes  a  limit  Avlicn  their  lunnbin'  n 
becomes  infinite,  the  sum  2/3,  =  ^^  +  /3.,  +  •  ■  •  +  /3„.  wliei'c  eacli  /?■  differs 
uniformly  from  the  corresponding  a-,  ly  an  infinitesimal  of  higher 
order,  will  approach  the  same  limit. 

As  a'  —  a  is  of  hitrher  order  than  a  and  j3'  —  ^  of  liiirlier  order  than  ;3, 

Ihn^^lzii^^O,       lim^::=^=.0       or       ^'  =  1  +  ^.       ^  =  1 -f  f , 
a  p  a  (3 

where  i;  and  f  are  infinitesimals.    Now  a'  =  cr  (1  -f  77)  and  /3'  =  (3(1  -I-  f).    Hence 

—  = ^     and     hm  —  =  hm  —  , 

a'      a  1  +  7]  a'  a 

provided  /3/a:  approaches  a  limit  ;  whereas  if  p/cx  becomes  infinite,  so  will  ji'/a'. 
In  a  more  complex  fraction  such  as  (/3  —  7)/«  it  is  not  permissible  to  replace  p 

*  It  caniKit  be  emphasized  too  stroTii^ly  tliat  in  tlic  syinliol  0/0  ilic  O's  arf  merely  sym- 
liolic  for  a  mode  of  variation  just  as  x  is:  they  ai-c  not  actual  O's  a:id  some  otlier  nota- 
tion Would  l>e  far  |)referable,  likewise  for  0  •  x,  0",  etc. 


64 


diffeiM':n tial  calculus 


and  7  individually  by  iuliuitesimals  of  higher  order;  for  /3  —  7  rnay  itself  be  of 
higher  order  than  /3  or  7.  Thus  tan  x  —  sin  x  is  an  inlinitesinial  of  the  third  order 
relative  to  x  although  tan  x  and  sin  x  are  only  of  the  first  order.  To  replace  tan  x 
and  sinx  by  infinitesimals  which  diffcu-  from  them  by  those  of  the  second  order  or 
even  of  the  third  order  would  generally  alter  the  limit  of  the  ratio  of  tanx  —  sinx 
to  x^  when  x  =  0. 

To  prove  Duhamers  Theorem  the  ^'s  may  be  written  in  the  form 

(3,- =  (l-;(l+  t;,-),  i  =  1,    2,    •  •  .,    Ji,  \-<]i\<e-, 

where  tlie  tj's  are  infinitesimals  and  wluu'e  all  the  t/'s  sinndtaneously  may  be  made 
less  tlian  the  assigned  e  owing  to  the  uniformity  r(M]uired  in  the  theorem.    Then 

I  (/3,  +  /3.,  +  •  •  •  +  fin)  -  {oc^  +  nr,  +  •  •  •  +  a,)  |  =  [  77,fr,  +  -q.^a.,  +  •  •  •  +  ■r]n^x„  \  <  eXa. 

Hence  the  sum  of  the  ^'s  may  be  made  to  differ  from  the  sum  of  thi^  a"s  by  less 
than  eSa,  a  (quantity  as  small  as  desired,  and  as  ^(v  approaches  a  limit  by  liypoth- 
esis,  so  2/3  nuist  approach  the  saim^  limit.  The  theorem  may  clearly  be  extended 
to  the  case  where  the  cr's  are  not  all  positive  provided  the  sum  S|a-i|  of  the  abso- 
lute values  of  the  a's  approaches  a  limit. 

35.    If  7/ =/'(.'•),  the  (I!ffrn;ntt,(fl  of  //  is  dcliiu'd  as 

<•///  =f'(,r)  A.r,  and  hence  (/.<■  =  1  ■  A,t.  (4) 

From  this  definition  of  (/>/  and  d.r  it  appears  tliat  (h//(J.r  =f'(.f),  wlieve 
tlui  quotient  (hj/dx  is  tiie  rpiotient  of  two  tinitt;  cpiautities  of  wdiicli  dx 
may  be  assigned  at  pleasure.  This  is  true  if  ,r  is  tlie  independent 
variable.    If  ./■  and  //  are  both  expressed  in  terms  of  t, 

x  =  x  {f),  y^ij  if),  ilx  =  I),x  (If,  (hj  =  ]),;/<n  ■ 

1/1/       1),)/ 


and 


(Ij 


/),> 


=  />.,//, 


by  virtue  of  (4),  §  2. 


From  this  a,p]")ears  i]]o,  impoi'tiint  theorem  :  77/7'.  (/iiotlcnt  di/fd.r  is  flip, 
dci'li'dflre  of  y  irilJi  rrsjicct  io  x  no  hniffcr  iiduit  Hu;  ln(h'p("nd('iif  rarhildc, 
111(11/  he.  It  is  this  theorem  which  I'eally  justilies  Avi'iting  the  dei'ivative 
as  a  fraction  ami  ti'eating  the  com})onent  differentials  according  to  the 
rules  of  ordinary  fi'actions.  For  higher  derivatives  this  is  not  so,  as 
may  be  seen  by  refei-ence  to  Ilx.  10. 

As  A//  a,nd  A./'  art;  I'egarded  as  infinitesimals  in  dctining  tlic  dcriva,- 
tivc,  it  is  natural  to  regard  d y  and  dx  as  infinitesimals.  The  ditferenc-e 
\y  —  dy  niay  be  pid'.  in  the  f'oi'm 

./•(•'■ +  A,r)-/60 


A//  —  «■///  = 


A./' 


-./"(■'■) 


A.r, 


(5) 


wherein  it  ap])ears  that,  when  A.r  =  0,  the  bra(l\et  a])])roa:clies  zero. 
Ilenee  arises  the  tlusorem  :  //'.'•  /x  ihc  ludcpcudcnf  (■((r'uildc  (oid  if  \y 
'ind  dy  (ire  ri'(i((rdcd  as  iiifnitcsi nials.  flic  dijf'cfc/icc  Ay —  df/  is  (oi  i/ijhl- 
ifcsiiiKil    of  /(i(/li('r   order   fhoii   \x.     4'liis    has    an    ap[)lication    fo    the 


TAYLOR'S  FORMULA;  ALLIED  TOPICS  65 

subject  of  cliange  of  variable  in  a  definite  integral.    For  if  x  —  <f>(t'), 
then  dx  =  (f>\f)df,  and  ap})arently 


f  f(x)dr=  fyicf^i^nwiO'^f, 


where  (fi(t^  =  (i  and  ^ (/,)  =  //,  so  that  t  ranges  from  t^  to  t^^  Avhen  x 
ranges  from  a  to  h. 

But  this  substitution  is  too  hasty ;  for  the  dx  written  in  the  integrand 
is  really  A.r,  Avhich  differs  from  dx  by  an  infinitesimal  of  higher  order 
when  X  is  not  the  independent  varial)le.  The  true  condition  may  be 
seen  by  comparing  the  two  siuns 

the  limits  of  which  are  the  two  integrals  a])o\e.  >i'ow  as  \x  dilTers 
from  dx  =  <f>'(i)dt  by  an  infinitesimal  of  higher  order,  so _/'(,>■) A,r  Avill 
differ  from /'[^(?')]  ^'(/*)c,V  by  an  infinitesimal  of  higher  order,  and 
with  the  proper  assumptions  as  to  continuity  the  difference  Avill  be  uni- 
form. Hence  if  the  infinitesimals /'(.>■)  A.r  be  all  positive,  Duhamel's 
Theoi'eni  may  l)e  ap})lied  to  justify  the  fornnda  for  change  of  variable. 
To  avoid  the  restriction  to  positive  infinitesimals  it  is  Avell  to  replace 
Duhamel's  Theorem  by  the  new 

THKoin::\i.  Osf/ond's  TJicnrcm.  Let  a^,  cr,,  •  •  •,  (x,^  be  n  infinitesimals 
and  let  «r,-  differ  unifoi'udy  by  infinitesimals  of  higher  order  than  A.r 
from   the   elements  /'(.'■,)  A,)\-  of   the   integrand   of  a  definite   integral 

Jf^x^dx,  where/'  is  continuous  ;  then  the  sum  1(t  --—  a^  +  '^.,  +  •  •  •  +  *■'„ 

a])proaches  tlie  value  of  the  definite  integral  as  a  limit  when  the  num- 
ber 71  becomes  infinite. 

Let  Hi  =./"(..■•,)  Ac, +  i',- A,'' ,■.  Nvlicri'  \^'i\  <e  owiuu:  t')  tlio  uniformity  deinaiuled. 

Then  I  2^<'V-  ^/{■ri)Ar,-\^--\^i';AJ-i\<e^AXi  =  e{h-a). 

But  as/ is  continuous,  the  delinite  inteii'ral  exists  and  one  can  make 


dj- 


€{b-a+l). 


^  /(.(•,•)  A.r,-  —  f  f{x)(li-    <  e,      and  hence         Vtr,;  -  f  '/(x) 

It  therefore  appears  that  2a/  may  be  made  to  differ  from  the  integral  In^  as  little 
as  desired,  and  Scr,-  nmst  then  approach  the  integral  as  a  limit.  Now  if  this  tlieo- 
rcm  be  applied  to  the  case  of  the  change  of  variable  and  if  it  be  assumed  that 
/[0(i)]  and  <p'{t)  are  continuous,  the  infinitesimals  A/,-  and  dxi  =  (p'{ti)dti  will 
differ  uniformly  (compare  Theorem  18  of  §  27  and  the  alxive  theorem  on  Ay  —  dij) 
by  an  infinitesimal  of  higher  order,  and  so  will  the  infinitesimals  /(x,)  A.rj  and 
f[(p  (ti)]  <p'{ti)  dti.  Hence  the  change  of  variable  suggested  by  the  hasty  substitution 
is  justified. 


66  DIFFERENTIAL  CALCULUS 

EXERCISES 

1.  Show  that  THospitars  Rule  applies  to  evaluating  the  indeterminate  form 
f{x)/<p{x)  -when  x  becomes  infinite  and  both /and  (p  either  become  zero  or  infinite. 

2.  Evaluate  the  following  forms  by  differentiation.  Examine  the  quotients 
for  left-hand  and  for  right-hand  approach  ;  sketch  the  graphs  in  the  neighborhood 
of  the  points. 

a^  —  //'■                       ,  ,     , .       tan  x  —  1  ,  ^   , . 

(a)    Inn ,  (/3)     hm ,  (7)   limxlogx, 

.-■  =  0         X  :r  =  ^Tr    X  —  \ir  x  =  0 

1 

(5)    limxc--'-,  (e)  lim(cotx)'*i°-%  (f)limxi-^. 

j=zo  .r  =  <)  0-  i  1 

3.  Evaluate  the  following  forms  by  the  method  of  expansions  : 

(1                         \                                        (>-r  (.Unx  loiTX 
cot-  X ) ,            (/3)  lim ,                            (7)  lim    ^^^ — , 
x"               I                   ,(■  =  0  X  —  tan  X                                  .(•  =  1 1  —  X 

,..,.,,  N  /  \  ,•      X  sin  (sinx)  —  sin^x  ,,.    ..      t-""— e-^— 2x 

(5)  lim  (cschx  —  cscx),        (e)  lim ^ j- ,        (f)  hm 

J  =  0  ^  =  0  x"  a;  =  0      X  —  sin  X 

4.  Evaluate  by  any  method  : 

,   ,   ,.      e^— e-' +  2sinx  —  4x                                      ,  ,  ,.      /tanx\^ 
(a)  hm ^—. ,  (/3)lim( )    , 

.TiO  X-^  a-  =  0\      X      / 

xcos-^r- log(l  4-x)  — siu-iix2  l(ig(x— ^tt) 

(7)  hm ^- — '- ^— ,  (5)     hm -—L, 

a-  =  o  x-^  .'■=>„■       tanx 


^^^iH'iH'+iJ-^^^^'K'+i)]- 


5.  Give  definitions  for  order  as  applied  to  infinites,  noting  that  higher  order 
would  mean  becDmiiig  infinite  to  a  greater  degree  just  as  it  means  becoming  zero 
to  a  greater  degree  for  infinitesimals.  State  and  prove  the  theorem  relative  to  quo- 
tients of  infinites  analogous  to  that  given  in  the  text  for  infinitesimals.  State  and 
prove  an  analogous  theorem  for  the  product  of  an  iiitinitcsimal  and  infinite. 

6.  Note  that  if  the  quotient  of  two  infinites  has  the  limit  1.  the  difference  of 
the  infinites  is  an  infinite  of  lower  order.  Apply  this  to  the  prriof  of  the  resolution 
ill  partial  fractions  of  the  (piotient/(x)/F(x)  of  two  polynoniials  in  case  the  roots 
of  the  denominator  are  all  real.  For  if  -F(x)  =  (x  —  (i)^'I-\{x).  the  (]uotient  is  an 
infinite  of  order  k  in  the  neighborhood  of  x  =  a  ;  but  the  difference  of  the  (luotient 
and/(a)/(x  —  (i)'-l-\{i()  will  be  of  lower  integral  order  —  and  so  on. 

7.  Sliow  that  when  x=-|-x.  the  function  e''  is  an  infinite  of  higher  order 
than  X"  no  matter  how  large  n.  Hence  show  that  if  I' {x)  is  any  polynomial, 
lim  P (/)<:-■''  =  0  when  x  =  +  x. 

8.  Show  that  (lou-  x)"*  when  x  is  infiinte  is  a  weaker  infinite  than  x"  110  matter 
how  large  »/  or  Imw  small  n,  supposed  positive,  may  be.  Wliat  is  the  graphical 
interpretation  ? 

9.  If  P  is  a  polynomial,  show  tliat  lim  P(-)c    ■'-  =  0.    Hence  show  that  the 


Maclaurin  development  of  c    •'-  is/(x)  =  c    ■'-  =  '    f("XOx)  if /(O)  is  defined  as  0. 


TAYLOR'S  FORMULA;  ALLIED  TOPICS  67 

10.  The  higher  differentials  are  defined  as  d>>y  =/<-") (^x)  (dr)"  where  x  is  taken 
as  tlie  independent  variable.  Show  that  d^'x  —  0  for  A;  >  1  if  x  is  the  independent 
variable.  Show  that  the  higher  derivatives  D^y,  D^y,  •  •  •  are  not  the  quotients 
d-y/dx-,  (Py/dx^,  •  •  •  \i  x  and  y  are  expressed  in  terms  of  a  tliird  variable,  but  that 
the  relations  are 

,    _  d-ydx  —  d-xdy  „    _dx  (dxd^y  —  dyd^x)  —  3  d-x  {dxd-y  —  dyd'-x) 

The  fact  that  the  quotient  d'^y/dx^,  n  >  1,  is  not  the  derivative  when  x  and  y  are 
expressed  paranietrically  militates  against  the  usefulness  of  the  higher  differentials 
and  emphasizes  the  advantage  of  working  with  derivatives.  The  notation  d"y/dx'^ 
is,  however,  used  for  the  derivative.  Nevertheless,  as  indicated  in  Exs.  10-19, 
higher  differentials  may  be  used  if  proper  care  is  exercised. 

11.  Compare  the  conception  of  higlier  differentials  witli  tlie  work  of  Ex.  5,  p.  48. 

12.  Show  that  in  a  circle  the  difference  between  an  infinitesimal  arc  and  its 
chord  is  of  the  third  order  relative  to  either  arc  or  chord. 

13.  Show  that  if  j3  is  of  the  }(th  order  with  respect  to  a.  and  y  is  of  the  first 
order  with  respect  to  ex,  then  (3  is  of  the  ?ith  order  with  respect  to  7. 

14.  Sliow  that  the  oi'der  of  a  product  of  infinitesimals  is  e(iual  to  the  sum  of  the 
orders  of  the  infinitesimals  when  all  are  referred  to  the  sameprimaiy  infinitesimal 
a.  Infer  that  in  a  product  each  infinitesimal  may  be  replaced  by  one  which  differs 
from  it  by  an  infinitesimal  of  higher  order  than  it  without  affecting  the  order  of  the 

product. 

15.  Let  A  and  B  be  two  points  of  a  unit  circle  and  let  the  angle  A  OB  subtended 
at  The  center  be  the  primary  infinitesimal.  Let  the  tangents  at  A  and  B  meet  at 
T,  and  OT  cut  the  chord  AB  in  ^^  and  the  arc  ^l  B  in  C.  Find  the  trigonometric 
expression  for  the  infinitesimal  difference  TC  —  CM  and  detennine  its  order. 

16.  Compute  d-  (x  sin  x)  =  (2  ens  x  —  x  sin  ,r)  dx"  +  (sin  x  +  x  cos  x)  d-x  by  taking 
the  tlifferential  of  the  differential.  Thus  find  the  second  derivative  nf  x  sin  x  if  x  is 
the  independent  variable  and  the  second  derivative  with  respect  to  t  if  x  =  \  +  t-. 

17.  Compute  the  first,  second,  and  third  differentials,  d-x  ^  0. 

(a)  j-cosx,  (/3)    Vl  —  X  log  (1  —  x),  (7)  /(.■'•'- sin  x. 

18.  In  Ex.  10  take  y  as  the  independent  variable  and  hence  express  d^pj.  l>^'y 
m  terms  of  DyX,  I)'^x.    Cf.  Ex.  10,  p.  U. 

19.  Make  the  changes  of  variable  in  Exs.  8.  0.  12.  p.  14.  by  tlie  metlmd  of 
differentials,  that  is,  by  replacing  the  derivatives  by  tin-  corresp(jnding  differential 
expressions  where  x  is  not  assiimed  as  independent  variable  and  Ijy  replacing  these 
differentials  by  their  values  in  terms  of  the  new  variables  where  the  higher  differ- 
entials of  the  new  independent  varialile  are  set  etjual  to  0. 

20.  Eeconsider  some  of  the  exercises  at  the  end  of  Chap.  I.  say,  17-10.  22.  23, 
27,  from  the  point  of  view  of  (Jsgood"s  Theorem  instead  of  the  Tlietjreni  of  the  Mean. 

21.  Find  the  areas  of  the  bounding  suifaces  of  the  solids  of  Ex.  11.  \).  18. 


68  DIFFERENTIAL  CALCULUS 

22.  Assume  the  law  F  =  kmm' / r-  uf  attraction  between  particles.  Find  the 
attraction  of  : 

(a)  a  circular  wire  of  radius  a  and  of  mass  3/ on  a  particle  m  at  a  distance  r  from 
the  center  of  the  wire  alon;,^  a  perpendicular  to  its  plane  ;      vl  n-s.  kMmr  (a-  +  r-)"  =. 

(^)  a  circular  disk,  etc.,  as  in  {a) ;  ^•l?i.s-.  2kM))i(i--(l  —  r/^r-  +  a-). 

(7)  a  semicircular  wire  on  a  jjarticle  at  its  center  ;  An^.  2kMni/Tra~. 

(6)  a  finite  rod  upon  a  particle  not  in  the  line  of  the  rod.  The  answer  should 
be  expressed  in  terms  of  the  an.i:le  the  rod  subtends  at  the  i)arti(le. 

(e)  two  parallel  equal  rods,  forming  the  opposite  sides  of  a  rectangle,  on  each 
other. 

23.  Compart-  the  method  of  derivatives  (§  7).  the  method  of  the  Theorem  of  the 
Mean  ($  17),  and  the  method  of  infinitesimals  above  as  applied  to  oljtaining  the  for- 
mulas for  ((f)  area  in  jiolar  coordinates,  (p)  mass  of  a  rod  oi  varial)le  density.  (7)  pres- 
sure on  a  vertical  suljmerged  bulkhead.  (0)  attraction  of  a  rod  on  a  i)article.  ( )btain 
the  results  by  each  method  and  state  which  method  seems  preferable  for  each  case. 

24.  Is  the  substitution  dx  =  (p'{t)dt  in  the  indt-linite  integral  //■(/)'J.f  to  obtain 
the  indefinite  integral    I  /[0(O]  <P'{i)<~lt  justifiable  immediately  ? 

36.  Infinitesimal  analysis.  To  work  rapidly  in  the  applications  of 
cali'ulus  to  })i-ol)lenis  in  g-eonietry  and  ])liysic.s  tmd  t(j  follow  readily  the 
books  "written  on  those  subjects,  it  is  necessary  to  have  scjine  familiarity 
with  working  directly  with  inhnitesinials.  It  is  ])ossil)le  ]»y  making  use 
of  the  Theorem  of  the  ]\Iean  and  allied  theorems  to  retain  in  every  ex- 
pression its  complete  exact  value  ;  but  if  that  expression  is  an  infini- 
tesimal which  is  ultimately  to  enter  into  a  ipiotieiit  or  a  limit  of  a  sum, 
any  infinitesimal  which  is  of  higher  order  than  that  which  is  tiltimately 
ke})t  will  not  influence  the  result  and  may  be  discarded  at  any  stage  of 
the  work  if  the  work  may  thereljy  be  simplified.  A  few  theorems 
worked  through  In*  the  infinitesimal  method  will  serve  }iartly  to  slnnv 
how  the  method  is  ttsed  and  ])artly  to  establish  I'esults  which  may  be 
of  use  in  further  work.    The  theorems  which  will  be  chosen  are  : 

1.  The  increntent  A./-  and  the  differential  '/.'■  of  a  varialjlc  differ  by 
an  infinitesimal  of  higher  order  than  either. 

2.  If  a  tangent  is  drawn  to  a  curve,  the  per])endicular  from  the  curve 
to  the  tangent  is  of  higher  order  than  the  distance  from  the  foot  of  the 
perpendicidar  to  the  ])oint  of  tangency. 

o.  An  infinitesimal  arc  differs  from  its  chord  liy  an  infinitt'simal  of 
higher  order  relative  to  the  arc. 

4.  If  one  angle  of  a  triangle,  none  of  whose  angles  are  infinitesimal, 
differs  infinitesimally  from  a  right  angle  and  if  //  is  the  side  o])]iosite 
ami  if  (^  is  am^ther  angle  of  the  triangle,  tlien  the  side  o]»])(isite  c^  is 
//  sin  c/)  except  for  an  infinitesimal  of  the  second  order  and  the  adjacent 
side  is  h  cos  ^  except  for  an  infinitesimal  of  the  first  order. 


TAYLOR'S  FOimULA;  ALLIED  TOPICb 


G9 


The  first  of  these  theorems  has  been  proved  in  §  35.  The  second  follows  from 
it  and  from  the  idea  of  tangency.  For  take  the  a:-axis  coincident  with  the  tangent 
or  parallel  to  it.  Then  the  perpendicular  is  Ay  and  the  distance  from  its  foot  to  tlie 
point  of  tangency  is  Ar.  The  quotient  Ay/Ax  approaches  0  as  its  limit  because  the 
tangent  is  horizontal  ;  and  the  theorem  is  proved.  The  theorem  would  remain  true 
if  the  perpendicular  were  replaced  by  a  line  making  a  constant  angle  iclth  the  tangent 
and  the  distance  from  the  point  of  tangency  to  the  foot  of  the  perpendicular  icere  re- 
placed by  the  distance  to  the  foot  of  the  oblique  line.    For  if  Z  PMX  —  0^ 

P/ 


PM 
TM 


PX  CSC 
TX^PX 


ji& 


PX 

fx 


CSC  ff 


1-^cot^ 
TX 


and  therefore  when  P  approaches  7'  with  0  cnnstant,  7' j// T.V  approaches  zero  and 
PM  is  of  higher  order  than  TM. 

The  third  theorem  follows  without  ditticulty  from  the  assumption  or  theoi'em 
that  the  arc  has  a  length  intermediate  between  that  of  the  chord  and  that  of  the 
sum  of  tlie  two  tangents  at  the  ends  of  the  chord.  Let  ^^  and  0.,  be  the  angles 
between  the  chord  and  the  taiiLrents.    Then 


s  -  A  n         A  r  +  Tr>  -  AB  _  a M (sec i^i  -  1)  +  37yi (sec  ^.,  -  1) 
AM  +  ZMn   ^      7l  M+JlB       "  AM  +  MJS 


(0) 


Now  as  AB  approaches  0,  both  sec  6*^  —  1  and  sec /9„  —  1  approach  0  and   their 

coefficients  remain  necessarily  finite.    Hence  tiie  difference  betwet'U  the  arc  and 

the  chord  is  an  infinitesinral  of  higher  onk-r  than  tlie  chord.    As 

the  arc  and  chord  are  therefore  of  the  same  order,  the  difference 

is  of  higher  order  than  the  arc.   This  residt  enables  (jne  to  replace 

the  arc  by  its  chord  and  vice  versa  in  discussing  infinitesimals  of 

the  first  order,  and  for  such  purposes  to  consider  an  infinitesimal 

arc  as  straight.    In  discussing  infinitesimals  of  the  second  ordei',  this  substitution 

would  not  be  permissible  except  in  vii-w  of  the  further  thcori-m  i;i\-en  below  in 

§  37,  and  even  then  the  substitution  will  hold  only  as  far  as  the  K-ngths  of  arcs  are 

concerned  and  not  in  regard  to  directions. 

For  the  fourth  theorem  let  &  be  the  angle  by  which  C  departs  from  'MP  and  with 
the  perpendicular  7i,V  as  radius  strike  an  arc  cutting  BC.    Then  by  trigonometry 

^1 C  =  A  .17  +  ^fr  =  h  cos  0  +  BM  tan  6, 
BC  =  h  sin0  +  7j'J7(sec  0  —  I). 

Now  tan  ff  is  an  infinitesimal  of  the  first  order  witli  respect  to  0  ; 
for  its  Maclaurin  development  l)egiiis  with  ff.  And  sec  6  —  1 
is  an  infinitesimal  of  the  second  order;  for  its  development 
begins  with  a  term  in  ff-.  The  theorem  is  therefore  proved. 
This  theorem  is  freiptently  applied  to  infinitesimal  triangles, 
that  is.  triangles  in  which  Ji  is  ti.>  approach  0. 

37-  As  a  further  discussion  of  the  third  theorem  it  may  be  recalled  that  by  deli- 
idtion  the  length  of  the  arc  of  a  curve  is  the  limit  of  the  length  of  an  inscribed 
polygon,  namely. 


:<i  C 


lim  (  ^  Ac; 


A7f  -'r  ^-  A/|  +  Ayj  +  •  •  •  +  ^  A/-;  +  Ay; 


70  DIFFEKEXTIAL  CALCULUS 

/ — :i — " 7,        rr', 7~^         ^'  +  ^y'  —  dx-  —  dy- 

2sow         V  Ax-  +  Ay-  —  y/dx-  +  ay-  =  — - 

VAj-  +  Ay-  +  V(Zj-  +  dy- 
_{Ar-  dj-)  (Ax  +  dx)  +  (Ay  -  d'y)  (Ay  +  dy) 


\  Ax-  +  Ay-  +  \  dx-  +  dy- 


and 


VAx-  +  Ay-  —  Vilx'^  +  dy-  _     (Ax  —  dx)  Ax  +  dx 


VAx-  +  Ay-  Vax-  +  Ay-  \  Ax-  +  Ay-  +  Vdx-  +  dy'^ 

i^y  -  dy) Ay  +  dy 

A  Ax-  +  Ay-  \  Ax-  +  Ay-  +  Vdx-  +  dy'^ 

Rut  Ax  —  dx  and  Ay  —  dy  are  infinitesimals  ijf  liiirher  order  tlian  Ax  ami  Ay. 
Hence  tlie  rii,dit-haiid  side  must  approai'h  zero  as  its  linnt  and  lienee  \  Ax-  +  Ay- 
differs  from  Vdx-  +  dy-  l)y  an  infinitesimal  of  hiulier  order  and  may  replaee  it  in 
the  sum 

s  =  lim  ^  ^  Ax,-  +  Ayr  =   lim  ^  Vdx-  +  dy-  =  C   '  Vl  +  y"-dx. 

The  length  of  the  are  measured  from  a  fixed  point  to  a  variable  point  is  a  func- 
tion of  the  tipper  limit  and  the  differential  of  arc  is 


df 


I  f   Vl  +  y'-dx  =  Vl  +  y'-dx  =Vdx-  +  dy-. 


To  find  the  order  of  the  difference  between  the  arc  and  its  chord  let  the  origin 
be  taken  at  the  initial  pctint  and  the  x-axis  tangent  to  the  curve  at  that  point. 
The  expansion  of  the  arc  by  Maclaurin"s  Formula  gives 

,s(x)  =  s(0)  +  j-.s'(0)  +  \  xH"{0)  +  1  x^s'"((9x). 

where         s (0)  =  0,         .s'(0)  =  Vl  +  y'-  o  =  L         ^"{^)  =  --— !    =  0. 

y  l-\-  y"-'io 

Owing  to  the  choice  of  axes,  the  expansion  of  the  curve  re<luces  to 

y  =/(■'■)  =  y  (•>)  +  .f//'(0)  +  I  x-y"{dx)  =  1  x-y'\&x), 

and  hence  the  chord  of  the  curve  is 


c  (x)  =  Vx--^  +  y-  =  X  Vl  +  1-  x-y  ((9x)]-  =  X  (1  +  x'-^P), 

where  P  is  a  complicated  expression  arising  in  the  exixvnsion  of  the  radical  by 
Maclaurin"s  Formula.    The  difference 

.s (X)  -  c  (X)  =  [X  +  1  x--s'"{dx)]  -  [X  (1  +  x'-^P)]  =  x^  (l  s"\e.r)  -  P). 

This  is  an  infinitesimal  of  at  least  the  third  order  relative  to  x.  Now  as  both  .s  (x) 
and  c  (x)  are  of  the  first  order  relative  to  x,  it  follows  that  the  difference  .s  (x)  —  r  (x) 
must  also  be  of  the  third  order  relative  to  either  .s-(x)  or  r(x).  Note  that  the  \)\\)oi 
assumes  that  y"  is  finite  at  the  point  considered.  This  result,  which  has  been 
found  analytically,  follows  more  simply  tlu.mgh  perhaps  less  rigorously  from  the 
fact  that  sec  <?!  —  1  and  sec  6.,  —  1  in  ((i)  are  infinitesimals  of  the  second  order  with 

e^  and  e... 

38.  Tlu!  tlicoiy  of  (■(iiifdcf  nf  jJnnp  c'lirrcs  may  1)0  treated  by  means 
(if  'rayloi'"s  l'~()rnnila  and  stated  in  terms  of  infinitesimals.  Let  two 
eurvt'S  1/ =  f{.rj  and  i/=(j(^.r^  be  tanyent  at  a  given  point  and  let  the 


TAYLOR'S  FORMULA;  ALLIED  TOPICS  71 

origin  be  c-hosen  at  that  point  with  the  a--axis  tangent  to  the  curves. 
The  Machuuin  developments  are 

y  =  A^-)  =  lf"(0),:^  +  •  •  •  +  ^^^^lyi  ^^-V'-'KO)  + 1 /'o/oo^O)  + . . . 
y=  .v(./0  =  ^rAO).'---=  +  -- ■ +  — ^ 

If  these  developments  agree  up  to  but  not  including  the  term  in  ./•",  the 
difference  between  the  ordinates  of  the  curves  is 

f(x)  -  g  (x)  =  ^ .."  [/•^")(0)  -  ;/"\0)]  +  •  •  • ,  /^«>(0)  ^  ./^''>(0), 

and  is  an  infinitesimal  of  the  nth  order  with  respect  to  .r.  The  curves 
are  then  said  to  have  fonttirt  of  order  n  —1  at  their  point  of  tangeney. 
In  general  when  two  curves  are  tangent,  the  derivatives  f"(0)  and  (/"(O) 
are  unequal  and  the  curves  have  simple  contact  or  rontorf  of  the,  Jirtit 
order. 

The  problem  may  be  stated  differently.  Let  PM  be  a  line  wliich 
makes  a  constant  angle  6  with  the  .''-axis.  Then,  Avhen  P  approaches  7', 
if  RQ  be  regarded  as  straight,  the  proportion 

lim  (PR  :  PQ)  =  lim  (sin  Z  PQR  :  sin  Z  PRQ)  =  sin  0  :  1 

shows  that  PR  and  PQ  are  of  the  same  order.  Clearly  also  the  lines 
TM  and  TX  are  of  the  same  order.    Hence  if 

PR  .        PQ 

lim =f^  0,  X,     then     lim ^  0,  x  . 

{TX)"         '      '  (TM)"         ' 


Hence  if  two  curves  have  contact  of  the  O;  —  l)st     "j^ 

order,  the  segment  of  a  line   intercepted   between  "^ 

the  two  curves  is  of  the  nth  order  with  res}»ect  to 

the  distance  from  the  point  of  tangeney  to  its  foot.    It  would  also  l)e 

of  the  ?ith  order  with  respect  to  the  pei-pendicular  TF  from  the  point 

of  tangeney  to  the  line. 

In  view  of  these  results  it  is  not  necessary  to  assume  tliat  the  two 
curves  have  a  special  relation  to  the  axis.  Let  two  curves  //  =  f(.r)  and 
1/  =  y  (,/■)  intersect  when  ./■  =  o^  and  assume  that  the  tangents  at  that  point 
are  ncjt  parallel  to  the  //-axis.    Then 

(,r^„)n-l  (;r  —  a)" 

!/  =  ih  +  (■'■  - ")/(")  +  •  •  •  +  - — Trr/'""'^('" )  +       ,     f'"H")  +  ■■■ 

{n—l)l  n.  ■ 

y  =  //o  -I- (■'■  - ") '/{")  +  ■■■  + T-rf"'-'H")  + r     ,'/"'< ")  +  ■■■ 

{/I  —1)  ,  n  . 


72  DIFFEKENTIAL  CALCULUS 

will  be  the  Taylor  developments  of  tlie  two  curves.  If  the  difference 
of  the  ordinates  for  equal  values  of  x  is  to  be  an  infinitesimal  of  the 
n\\\.  order  wnth  respect  to  x  —  a  which  is  the  perpendicular  from  the 
j^oint  of  tangency  to  the  ordinate,  then  the  Taylor  developments  must 
agree  up  to  but  not  including  the  terms  in  x'\  This  is  the  condition  for 
contact  of  order  n  —  1. 

As  the  difference  between  the  ordinates  is 

/(■'■)  -  y  (■'■)  =  ^  (•«  -  '0"  [/'"K")  -  ^^"^(^0]  +  ■■■, 

the  difference  will  (-hange  sign  or  keep  its  sign  when  x  passes  through 
((  according  as  7i  is  odd  or  even,  because  for  values  sufficiently  near  to 
./•  the  higher  terms  may  be  neglected.  Hence  the  curves  tclll  cross  each 
other  if  the  order  of  contact  is  even^  but  will  not  cross  each  other  if  the 
order  of  contact  is  odd.  If  the  values  of  the  ordinates  are  eiiuated  to  find 
the  points  of  intersection  of  the  tAvo  curves,  the  result  is 

0=  l(.r- a)"  ^  [/<")(.) -,V^"Y/0]  +  ---^ 

and  shows  that  x  =  a  is  a  root  of  multiplicity  w.  Hence  it  is  said  that 
two  curves  have  in  common  as  many  coincident  points  as  the  order  of 
their  contact  plus  one.  This  fact  is  usually  stated  more  graphically 
1)V  saying  that  t/ie  ci/rres  hare  n  consecutire  p'>ints  in  coDnnon.  It  may 
be  remarked  that  what  Taylor's  development  carried  to  n  terms  does,  is 
t(j  give  a  polynomiid  which  has  contact  of  order  w  —  1  with  the  function 
that  is  developed  by  it. 

As  a  problem  on  contact  consider  the  determination  of  tlie  circle  which  shall 
liave  contact  of  the  second  order  with  a  curve  at  a  given  point  (a,  yo).    Let 

y  =  yo  +  (.r  -  '0/'(")  +  2  (•''  -  «)'/"(")  +  •  ■  • 
be  the  development  of  the  curve  and  let  y'  =f\a)  =  tan  r  lie  the  slope.    If  the 
circle  is  to  have  contact  with  the  curve,  its  center  nuist  be  at  some  point  of  the 
normal.    Then  if  It  denotes  the  assumed  radius,  the  equation  of  the  circle  may  be 
written  as 

(./•  -  a)-  +  2  n  sin  t  (,c  -  a)  +  (//  -  //o)"  -  2  /.'  cos  r  (//  -  //„)  =  0, 

where  it  remains  to  detei'mine  7i'  so  that  tlie  deveUipnient  of  the  circle  will  coincide 
with  tliat  of  the  curve  as  far  as  written.    Differentiate  the  eciuation  of  the  circle. 

ily        11  sin  T  +  (,/■  —  a)  /(h/\ 


=  tan  T  =  /''(((), 
(/,/■        7.' cos  T  —  (// —  v/J  \d.C/a,!,„ 

(V^y  __  [ //  c. .s T  -  (//  -  y„)Y-  +  [ II ^T  +  (x-ji)Y  ld'y\ 1_ 

and  y  =-^  ?/„  +  (./■  -  ")/'{'()  +  i  {f  -  a)-  --  -,-  +  •  •  • 

/i   COS"  T 


TAYLOirS  FOILAIULA:  ALLIED  TOPICS 


is  the  development  of  the  circle.    The  equation  of  the  coethcieiit.s  of  (/  —  a)-, 

=  /     <0,     ii'ives     R  = =  - —     '--    '-':'-^  . 

Z^co.s'r     •     ^  "     -  /"(„)  r(a) 

This  is  the  well  known  fdrnuila  for  the  radius  of  curvature  and  shows  that  the  cir- 
cle of  curvature  has  contact  of  at  least  the  second  order  with  the  curve.  The  circle 
is  sometimes  called  the  osculatiuic  circle  instead  of  the  circle  of  curvature. 

39.  Three  theorems,  one  in  geometry  and  two  in  kinematics,  will 
now  Ije  proved  to  illustrate  the  direct  application  of  the  infinitesimal 
methods  to  such  problems.    The  choice  Avill  be  : 

1.  The  tangent  to  the  ellipse  is  eqttally  inclined  io  the  focal  radii 
drawn  to  the  point  of  contact. 

2.  The  displacement  of  any  rigid  body  in  a  plane  may  be  regarded 
at  any  instant  as  a  rotation  through  an  infinitesimal  angle  about  some 
point  unless  the  body  is  moving  parallel  to  itself. 

3.  The  motion  of  a  rigid  body  in  a  i)lane  may  be  regarded  as  the 
rolling  of  one  curve  upon  another. 

F(.)r  the  first  prohlem  consider  a  secant  J'l"  which  may  he  convi'rted  into  a 
tan,<:ent  7'7''  l)y  letting;  the  tw()  points  approach  until  they  C(_iincii]e.  Draw  the 
focal  radii  to  Z'  and  P'  and  strike  arcs  with  /•"  and  F'  as 
centers.  As  F'P  +  PF  :=  FT'  +  P' F  ^  2  a.  it  follows 
that  XI'  =  MP'.  Now  consider  the  two  triangles  PP'M 
and  P'PX  nearly  rii:ht-ang-led  at  3/  and  X.  The  sides 
PP\  I'M.  PX.  P'M.  P'X  are  all  infinitesimals  of  the 
same  order  and  of  the  same  order  as  the  angles  at  /•'  and 
F'.    Hy  proposition  -t  of  §  ?A\ 

MP'  =  PP'  cos  Z  PP'M  +  f  J.  XP  =  PP'  (;<  is  Z  P'J'X  +  c.„ 

where  c,  and  c,  are  intinitesimals  relative  to  MP'  and  XP  or  /'/".    Therefon 


lim  [cu>ZPP'M-  cosZ  7"7'-V]  =  cos  Z  TPF -  cos  Z  T  PF'  =  lim 


PP 


0. 


:>7B' 


and  the  tw()  anples  TPF'  and  T'PFixyc  proved  to  Ije  eijual  as  desireil. 

To  prove  the  second  the(U'em  note  first  that  if  a  body  is  riuiil.  its  ]iositioii  is  i-om- 
pletely  deternnned  when  the  position  ^l  B  of  any  rectilinear  seL:inent  of  the  hody 
is  known.  Let  the  points  .1  and  P  of  the  liody  be  de- 
scribini:-  curves  ^1^-1' and  BW  so  that,  in  an  infinitesimal 
interval  of  time,  the  line  ^4 7^  takes  the  neiizhborinir  posi- 
tion J  '7i'.  Krcct  tlie  perpemlicular  liisectoj-s  of  the  lines 
.1.1'  and  BJV  ami  let  them  intersect  at  O.  Then  the  tri- 
auLiles  .107i  and  A'OJV  have  the  three  sides  of  the  one 
eijual  to  the  three  sides  of  the  other  and  are  equal,  and 
the  second  may  be  oljtained  from  tlie  lirst  liy  a  mere  rotation  about  C)  throuLLii  the 
an-le.l0.r=  BOB'.  Except  for  intinitesimals  of  higher  order,  the  ma-nitude  of 
the  aimle  is  AA'/OA  uv  BB'/Oll.  Next  let  the  interval  of  time  approach  0  so  that 
.1 '  appi-oaclies  A  and  P>'  approaches  B.    The  perpendicular  liisi/ctors  w\\\  approach 


74  DIFFERENTIAL  CALCULUS 

the  normals  to  the  arcs  AA'  and  BIV  at  A  and  7i,  and  the  point  0  will  approach 
the  intersection  of  those  normals. 

The  theorem  may  then  be  stated  that:  At  any  instant  of  time  the  motion  of  a 
rigid  body  in  a  plane  may  he  considered  as  a  rotation  through  an  infinitesimal  angle 
about  the  intersection  of  the  normals  to  the  paths  of  any  two  of  its  points  at  that  in- 
stant ;  the  amount  of  the  rotation  will  be  the  distance  ds  that  any  point  moves  divided 
by  the  distance  of  that  point  from  the  instantaneous  center  of  rotation;  the  angular 
velocity  about  the  instantaneous  center  icill  be  this  amount  of  rotation  divided  by  tJte 
interval  of  time  dt,  that  is,  it  loill  be  r/r,  where  v  is  the  velocity  of  loiy  poiid  of  tJie  body 
and  r  is  its  distance  from  the  instantaneous  center  of  rotation.  It  is  therefcn-e  seen 
that  not  only  is  the  desired  theorem  proved,  bnt  numerous  other  details  are  fnund. 
As  has  been  stated,  the  point  about  which  the  body  is  rotating;  at  a  givi-ii  instant 
is  called  the  instantaneous  center  for  that  instant. 

As  time  goes  on,  the  position  of  the  instantaneous  center  will  generally  change. 
If  at  each  instant  of  time  the  position  of  the  center  is  marked  on  the  moving  plane 
or  body,  there  results  a  locus  which  is  called  the  moving  centrode  or  body  centrode ; 
if  at  each  instant  the  position  of  the  center  is  also  marked  on  a  fixed  plane  over 
wliich  the  moving  plane  may  be  considered  to  glide,  there  results  another  locus  which 
is  called  the  fixed  centrode  or  the  sjjace  centrode.  From  these  (lefinitii)ns  it  follows 
that  at  each  instant  of  time  the  body  centrode  and  the  space  centrode  intersect  at 
the  instantaneous  ct'Uter  for  that  instant.  Consider  a  series  of 
po.sitions  of  the  instantaneous  center  as  r_.J'_il']\P.^  marked 
in  space  and  C^_or^;_iQQjQ.,  marked  in  the  l)o(ly.  At  a  given 
instant  two  of  the  points,  say  P  and  Q,  coincide  ;  an  instant 
later  the  body  will  have  moved  so  as  to  bring  Q^  into  coin- 
cidence with  P,  ;  at  an  earlier  instant  Q_i  was  coincident  with 
P_i.  Now  as  tlie  motion  at  the  instant  when  P  and  Q  are  together  is  one  of 
rotation  through  an  infinitesimal  angle  about  that  point,  the  angle  between  PP^ 
and  QQ^  is  infinitesimal  and  the  lengths  PP^  and  QQj  are  efjual  ;  for  it  is  by  the 
rotation  about  P  and  Q  that  (^^  is  to  be  l)rouglit  into  t'oincidence  with  7^  Hence 
it  follows  r^  that  the  two  centrodes  are  tangent  and  2^  that  the  distances  7'/'^  =  (^Qj 
which  the  point  of  contact  moves  along  the  two  curves  during  an  infinitesimal  inter- 
val of  time  are  the  same,  and  this  means  that  the  two  curves  roll  on  one  another 
without  slipping  —  l)ecause  the  veiy  idea  of  slipping  implies  that  the  point  of  con- 
tact of  the  two  curves  should  move  by  different  amounts  along  the  two  curves, 
the  difference  in  the  amounts  being  the  amount  of  the  slip.  The  third  theorem 
is  therefore  proved. 

EXERCISES 

1.  If  a  finite  parallelogram  is  nearly  rectangied.  what  is  the  order  of  infinites- 
imals neglected  by  taking  the  area  as  the  ijroduet  of  the  two  sides'.'  Wliat  if  the 
figure  were  an  isosceles  trapezoid?  Wiiat  if  it  were  any  rectilinear  ((iiadrilateral 
all  of  whose  angles  differ  from  right  angles  by  infinitesimals  of  the  same  oi-der '.' 

2.  On  a  sphere  of  radius  /•  the  area  of  the  zone  between  the  parallels  of  latitude 
X  and  \  -t-  '/\  is  taken  as  2  ttc  eos  \  •  nl\.  the  i)erimeter  of  the  base  times  the  slant 
height.  ( >f  what  order  relative  to  d\  is  the  infinitesimal  neiileeted  ?  What  if  the 
perimeter  uf  tlie  middle  latitude  were  taken  so  that  2  tt/'- cos  (\  +  \d\)d\  were 
assumed  ".' 


TAYLOR'S  F()E:\[ULA:  ALLIED  TOPICS  To 

3.  What  is  tlie  order  of  the  iiifinitesinia]  iiedected  in  taking  i-Trr-dr  as  the 
vohune  of  a  hollow  sphere  of  interior  railius  r  and  th.iekness  dr  ?  What  if  the  mean 
radius  were  taken  instead  of  the  interior  radius  '?  ^^'l>uld  any  particular  radius  be 
best  ? 

4.  Discuss  the  length  of  a  space  curve  y  =f(j).  z  =  g  {x)  analj'tically  as  the 
length  of  the  plane  curve  was  discussed  in  the  text. 

5.  Discuss  proposition  2,  p.  08.  by  Maelaurin's  Formida  and  in  particular  show 
that  if  the  second  derivative  is  continui)Us  at  the  point  of  tangency.  the  infinites- 
imal in  t^uestion  is  of  the  second  order  at  least.    How  a.ltout  the  case  of  the  tractrix 


a . 


y  =  ~r  log +  ^  "-  -  /-, 

^        a  +  \a-  -  X- 

and  its  tangent  at  the  vertex  x  =  a?    How  alxiut  s{x}  —  c(.r)  of  §  37  ? 

6.  Show  that  if  two  curves  have  contact  of  order  ?i  —1,  their  derivatives  will 
have  contact  of  order  n  —  2.  What  is  the  order  of  contact  of  the  A'th  derivatives 
k<n-l? 

7.  State  tlie  conditions  for  maxima,  minima,  ami  points  ot  int^ection  in  the 
neighborhood  of  a  point  where /'")((/)  is  the  first  derivative  that  does  not  vanish. 

8.  Determine  the  order  of  contact  of  these  curves  at  their  intersections: 

9.  Show  that  at  points  where  the  radius  of  curvature  is  a  maximum  or  mini- 
nuun  the  contact  of  the  osrulating  circle  with  the  curve  nuist  be  of  at  least  the 
third  order  and  must  always  Ije  of  odd  order. 

10.  Let  7'.V  lie  a  normal  to  a  curve  and  F'X  a  neighborinir  normal.  If  O  is  the 
center  of  the  osculatini:  circle  at  P.  show  with  the  aid  of  Kx.  0  tiiat  ordinarily  the 
perpendicular  from  O  to  P'X  is  of  tlie  second  order  relative  to  the  arc  PI''  and  that 
the  distance  OX  is  of  the  first  ordi-r.  HeiK/e  interpret  the  statement  :  Consecutive 
normals  to  a  curve  meet  at  the  center  of  the  osculating  circle. 

11.  Does  the  osculating  circle  cross  the  curve  at  the  point  of  osculation  '.'  Will 
the  osculating  circh.-s  at  neiglil)oriiig  points  of  the  curve  intersect  in  real  points? 

12.  In  tlie  hyperbola  the  focal  radii  drawn  to  any  point  make  equal  angles  with 
the  tangent.    Trove  this  and  state  and  prove  the  corresponding  theorem  for  the 

parabola. 

13.  Given  an  infinitesimal  arc  AB  cut  at  C  by  the  perpendicular  bisector  of  its 
chord  AB.    What  is  the  order  of  the  difference  AC  —  BC  ? 

14.  of  what  order  is  the  area  of  the  segment  included  between  an  infinitesimal 
arc  and  its  chord  compared  with  the  sijuare  on  the  chord  ? 

15.  Two  sides  AB.  AC  of  a  triangle  are  finite  and  differ  infinitesimally  :  the 
aiiirle  6  at  *1  is  an  infinitesimal  of  the  same  order  and  the  side  BC  is  either  recti- 
linear or  curvilinear.  What  is  the  order  of  the  neglected  infinitesimal  if  the  area 
is  assumed  as  \  AB'O  ?    What  if  the  assumption  is  5  AB  ■  AC  ■  B  ? 


76  DIFFERENTIAL  CALCULUS 

16.  A  cj'cloid  is  the  locus  of  a  fixed  point  upon  a  circumferenre  whicli  rolls  on 
a  straight  line.  Show  that  the  tangent  and  normal  to  the  cycloid  pass  through  the 
highest  and  lowest  points  of  the  rolling  circle  at  each  of  its  instantaneous  positions. 

17.  Show  that  the  increment  of  arc  Ah  in  the  cycloid  differs  from  2o  sin  I  6d0 
by  an  infinitesimal  of  higher  order  and  that  the  increment  of  area  (betweeii  two 
consecutive  normals)  differs  from  3  a-  sin-  I  BdO  by  an  infinitesimal  of  higher  order. 
Hence  show  that  the  total  length  and  area  arc  8«  and  i-yira-.  Here  a  is  the  radius 
of  the  generating  circle  and  6  is  the  angle  subtended  at  the  center  by  tlie  lowest 
point  and  the  fixed  point  which  traces  the  cycloid. 

18.  Show  that  the  radius  of  curvature  of  tlie  cycloid  is  bisected  at  the  lowest 
point  of  the  generating  circle  and  hence  is  4  a  sin  I  9. 

19.  A  triangle  ABC  is  circumscri!)ed  al)out  any  nval  curve.  Show  that  if  tlie 
side  BC  is  bisected  at  the  p(untof  contact,  the  area  of  tlie  triangle  will  b(.'  changed 
by  an  infinitesimal  of  the  second  order  when  BC  is  replaced  l.)y  a  neighboring  tan- 
gent B'C\  Vnit  that  if  BC  be  not  Ijiseeted,  the  change  will  Ije  of  the  first  order. 
Hence  infer  that  tlie  minimum  triangle  circumscribed  aljout  an  oval  v.ill  have  its 
three  sides  bisected  at  the  points  of  contact. 

20.  If  a  string  is  wrapped  alxnit  a  circle  f)f  radius  n  and  then  unwound  so  that 
its  end  describes  a  curve,  show  that  the  lengtli  of  the  curve  and  the  area  between 
the  curve,  the  circle,  and  the  striiiL:-  are  'x'\ 

H=  /    oMO,  A  =  I     \<(-d-(W.  ^     V 

where  0  is  the  angle  that  the  luiwinding  string  has  tui-ned  through. 

21.  Show  that  the  motion  in  space  of  a  rigid  body  one  point  of  wliich  is  fixed 
may  be  regarded  as  an  instantaneous  rotation  about  some  axis  thrnugh  the  gi\'en 
point.  To  do  this  examine  the  displacements  of  a  unit  sphere  surrounding  tlie  lixed 
point  as  center. 

22.  vSuppose  a  fluid  of  \arialile  <leii>ity  7^(.;')  is  tinwiiiL;'  at  a  uivcn  instant  through 
a  tube  surrounding  the  .r-axis.  Let  the  velocity  of  the  fiuid  he  a  function  r(,r)  <if  ,/'. 
Show  that  during  the  intiiiitesimal  time  ot  the  diminution  of  the  amount  of  liie 
Ikiid  which  lies  betweeJi  ./'  =  a  and  ./:  =  a  +  h  is 

.S  [v  (a  +  h)  I)  [a  +  //)  U  -  V  (r/)  I)  (n)  51], 

where  .S  is  tlie  cross  sectir)n  of  the  tube.    Hence  show  that  T)(.r)  r(,r)  =  const,  is  the 
condition  that  the  flow  of  the  fiuid  shall  n(_)t  change  tlie  dcii.-ity  at  any  point. 

23.  Consider  the  curve  u  =./'(■'■)  and  three  equally  s['afed  ordinatcs  at  ,/•  ~  a  —  3. 
J-  =:.  (I.  X  =  II  +  S.  Inscribe  a  trajiczoid  by  joinini:-  the  emls  of  the  ordiiiati-s  at 
,/•  =  II  -±  S  and  circums(;ri!ic  a  trajiezoid  by  di'awing  the  tangent  at  tlie  end  of  the 
ordinate  at  .r  =  u  and  prodiicing  to  mei't  the  other  onlinates.    Siiow  that 

•S  =  2  of{ii),  S  =  2  o\f(ii)  +  "'-  f"(ii)  +    ''  -  ./"("-'(t)  I ' 

L  'i  120  J 

r  -      «"■-  ■,.      5^  .      1 

Nj  =  2  5    ,t{<i)  -r   _^  .fill)  J^  _^  y^'l^'i) 


TAYLOR'S  FORMULA;  ALLIED  TOPICS  77 

are  the  areas  of  the  circumscribed  trapezoid,  the  curve,  tlie  inscriljetl  trapezoid. 
Hence  infer  that  to  compute  the  area  under  the  curve  from  tlie  inscribed  or  cir- 
cumscribed trapezoids  introduces  a  relative  error  of  the  order  5-,  but  that  to  cf)m- 
pute  from  tlie  relation  6'  =  i  (2  S^^  +  .S\)  introduces  an  error  of  oidy  the  order  of  5*. 

24.  Let  the  interval  from  a  to  h  be  divided  into  an  even  niunber  2n  of  eipial 
parts  5  and  let  the  2  n  +  1  ordinates  (/q,  y^,  •  •  •,  y-2n  at  the  extremities  of  the  inter- 
vals be  drawn  to  the  curve  y  =/(j).  Inscribe  trapezoids  by  joininsj;  the  ends  of 
every  other  ordinate  beginning  with  y^,  y„,  and  going  to  y-^n-  Circumscribe  trape- 
zoids by  drawing  tangents  at  the  ends  of  every  other  ordinate  ?/j,  y.,,  ■  ■  •,  yon-i. 
Compute  the  area  under  the  curve  as 

.S=f'f(,r)dx  =  -~~  [4  0/,  +  ?/,  +  ---  +  U-2n-i) 

+  2  (;/„  -I-  v/.,  -I 1-  y2„]  -  //o  -  1/2 „]  +  li 

liy  using  IIk'  work  of  Ex.  23  and  infer  that  the  error  R  is  less  than  {h—a)  S'*f<-'^'>{^)/4:0. 
This  method  of  computation  is  known  as  Shupson's  Ilulc.  It  usually  gives  accu- 
racy sufficient  for  work  to  four  or  even  live  tigures  when  5  =  0.1  and  b  —  a  ~  I  ;  for 
/<"'){j')  usually  is  small. 

25.  Compute  these  integrals  by  Simpson's  Kule.    Take  2n—  10  equal  intervals. 
■     Carry  numerical  work  to  six  tigures  except  where  tables  nuist  be  used  to  lind/'(,r)  : 

la)    r "  ~  =  lotr  2  =  0.G0.315,  (^    f  ^  — ~~  =  tan-i  1  =  ^  tt  =  0.78535, 

Jl        X  '~  J<)       I   -I-  X-  i 

(7)    f  '    sin  x(U  =  1.00000,  (5)    f '  log,„.fr7x  =  2  log,,j  J"  -  ^t  =  0.16770, 

Jo  Jl 

(0    r'^^^^+^.Lr  =  0.27220,  (,')    f  ^^±^  dx  =  0.^2247. 

Jo         1  +  X-  Jo  X 


1  log  n_+.r),,  _,.,„....,  ,,,    ^Mog(l  +  .f) 

+  x 

The  answers  here  given  are  the  true  values  of  (he  integi'als  to  live  places 


26.  Show  that  the  (juailrant  of  tln'  ellipse  x  =  u  slw  cp,  y  =--.  hntscp  is 

/■  A  TT    , ; n  I     , 

,s  =  (f  /        \  1  —  c-  sin-  <pd<p  =  ,',  TTd  I     ^  .1  (2  —  t-)  -i-  ,\  c-  cds  7r«  ilu. 

Jo  '       Jo        ' 

Compute  to  four  figures  h\  Simpson's  Kule  with  six  divisions  the  quadrants  of 
the  ellipses  : 

(a)  c=\  \/3,     s  =  1.211  a,  (/S)  c  =  \  V2,     h  =  1 .351  a. 

27.  Expand  s  in  Ex.  26  into  a  series  and  discuss  the  remainder. 

1        /I  •  3-  •  •  ("^  ?)  -I-  lU-  e-"  +  - 

/.'„  < (  -     -    ~'\ SeeEx.l8.p.60.andPeirce-s"Tables.'-p.62. 

1_  e--;  \2.4-..(2?i+ 2)/ 2n-|-l 

Estimate  the  number  of  terms  necessary  to  ccmipute  Ex.  26  (/3)  with  an  error  not 
greater  than  2  in  the  last  place  and  compare  the  labor  with  that  of  Simpson's  Rule. 

28.  If  the  eccentricity  of  an  ellipse  is  ,,L.  find  to  five  decimals  the  percentage 
error  made  in  taking  2 7r«  as  the  perimeter.  ,  An>i.  0.006i>i% 


78 


DIFFEKEXTIAL  CALCULUS 


29.  If  tlic  catenary  y  ~  r  cosli  {.c/r)  .i,dves  the  sliape  of  a  wire  of  lensth  L  sus- 
peiiiled  between  two  points  at  the  same  level  and  at  a  distance  I  nearly  equal  to 
L.  iind  the  first  approximation  coiniectini;  L.  I.  and  d,  where  d  is  the  dip  of  the 
wire  at  its  lowest  point  below  tlie  level  of  support. 

30.  At  its  middle  point  the  paraliolic  cable  of  a  .suspension  bridge  1000  ft.  long 
between  the  supports  sags  50  ft.  below  the  level  of  the  ends.  Find  the  length  of 
the  ealile  correct  to  inches. 

40.  Some  differential  geometry.  Suppose  that  Ijetween  the  incre- 
ments (»f  a  set  of  varial)]os  all  of  which  depend  on  a  single  varial)le  t 
tlu'i'o  exists  an  equation  which  is  true  except  for  infinitesimals  of  higher 
order  than  \t  —  dt,  then  the  equation  will  Ia'  exactly  true  for  the  differ- 
entials of  the  variables.    Thus  if 

f\x  +  ;/Ay  +  /,  A-  +  /A/  +  •  •  •  +  .>^  +  .',  +  . . .  =  0 

is  an  e([uation  of  the  sort  mentioned  and  if  the  coefficients  are  any  func- 
tions of  the  variables  and  if  c^  r.„  •  ■  •  are  infinitesimals  of  higher  order 
than  dt,  the  limit  of 


,A,/-  A//        ,  A.v        ,\t 

''  \t       ■'  \t  \t         \f 


A;'       A^ 


IS 


d.r  <hl  dz 

or  fd.r  +  </d,j  +  /,,/,-:  +  /r/^  =  0 : 

and   the   statement  is  proved.    This    I'csult  is   very  ttseful   in   wi-iting 

down  vai'ious  differential  formulas  of  geometry  where  the  a}»])roximate 

relation   between  the  increments  is  obvious  and  where  the  true  relation 

between  tlie  differentials  can  therefore  be  found. 

]""or  instance  in  the  case  of  the  differential  of  ai'c  in  rectangular  cocu'- 

dinates.  if  the  increment  of  arc  is  known  to  differ  from  its  chord  by  an 

infinitesimal  of  higher  order,  the  Pythagorean  theorem  shows  that  the 

equation  .2       v    -2   ,    \  ,-2     ,„.      v  2       \  .2  ,    *   ,2   ,    k  ..-i  /~\ 

^  A.s    =  A.'    +  A//        or      An    =  A.'    +  A//   -f-  A.;  (^  i  j 

is  true  except  for  infinitesimals  of  higher  order:   and  hence 

ds-  =  d.r-  +  dir       or       ds-  =  d.r-  +  dir-\-dz-.  (7') 

In    the  case   of  jilane   jiolar   coin-dinates,  the   triangle  PP'X  (see   Fig.) 

has  two  curvilinear  sides  /'/•■'  and  /'.V  and  is  right-  Jr 

angled    at   X.     The    Pythagorean    theorem    may  be 

apj)lied  to  a  curvilinear  triangle,  or  the  tiiangle  may 

be  replaced  by  the  I'cctilinear  triangle  PP'X  with  J.r" 

the  angle  at  A'  no  longer  a  right  angle  but  nearly  so.    In  either  way  of 

looking  at  the  figui-e,  it  is  t'asily  seen  tliat  the  eipiation  Ax-  =  A/'-  +  r^(f>- 


TAYLOR'S  FORMULA;  ALLIED  TOPICS 


79 


which  the  figure  suggests  differs  from  a  true  equation  by  an  infinitesi- 
mal of  higher  order;  and  hence  the  inference  that  in  polar  coordinates 
ds'  =  d)^  -f-  i^d(^'. 

The  two  most  used  systems  of  coordinates 
other  than  rectangular  in  si)ace  are  the  pohw 
or  spherical  and  the  ojl'mdrical .  In  the  first 
the  distance  r  =  OP  from  the  pole  or  center, 
the  longitude  or  meriditmal  angle  </>,  and  the 
colatitude  or  polar  angle  6  are  chosen  as  coor- 
dinates;  in  the  second,  ordinary  polar  coordinates  r  =  OM  and  (f>  in 
the  a-y-plane  are  coml>ined  with  the  ordinary  rectangular  ;:  for  distance 
from  that  plane.    The  formulas  of  transformation  are 

z  —  r  cos  6, 


=  Vj'-  -f-  /  +  .t'> 


7/  =  /•  sin  6  sin  <^, 
X  =  r  sin  6  cos  <f>, 


V?  +  y^  +  .^' 


(8) 


(j)  =  tan" 


(9) 


for  polar  coordinates,  and  for  cylindrical  coordinates  they  are 
z  =  z,     u  =  r  sui  ({>.     ./•  =  /•  cos  (/),     r —\  J- -\- >/-,     </>  — tan"^-- 

X 

Formulas  such  as  that 
for  the  differential  of 
arc  may  l)e  obtained  for 
these  new  co(')rdinates  l)y 
mere  transformation  of 
(7')  according  to  the  rules 
for  change  of  variable. 

In  l}oth  these  cases, 
however,  the  value  of 
ds  may  be  found  readily 
by  direct  inspection  of 
the  figure.  The  small 
parallelepiped  (figure 
for  polar  case)  of  which 
As  is  the  diagonal  has 
some  of  its  edges  and 
faces  curved  instead  of 
straight;  all  the  angles, 
however,  are  right  angles, 
and  as  the  edges  are  infinitesimal,  the  equations  certainly  suggested  as 
holding  except  for  infinitesimals  of  higher  order  are 


80  DIFFKKEXTIAIi  CALCULUS 

As"-^  =  Ar^  +  r  shr  OAc{>-  +  rAd-     and     A.s-  =  A/--  +  /'-Ac^-  +  A.-r    (10) 
or    (lr  =  (/r  +  rshre-/(f>--{-r-'/0'       and      r/.s-- =  r/y- + /■'-^/^'- +  r/--.     (10') 

To  make  the  })roof  (•oiii})U-t(',  it  would  be  iiecessaiy  to  sliow  tliat  iiotli- 
ing  but  intinit(,\sinials  of  higher  order  liave  been  neglected  and  it  might 
actually  be  easier  to  transfoi'ni  V'/,/'- +  '///-  + r/--  rather  than  give  a 
rigorous  demonstration  of  this  fact.  Indeed  the  iniinitesinial  method  is 
seldom  iised  rigorously;  its  great  use  is  to  make  the  facts  so  clear  to  the 
rapid  worker  that  he  is  Avilling  to  take  the  evidence  and  omit  the  proof. 
In  the  i)lane  for  rectangular  coiu'dinates  with  ndings  parallel  to  the 
_y-axis  and  for  polar  coordinates  with  rulings  issuing  from  the  pole  the 
increments  of  area  differ  from 

JA=;/J.r     and     (/A  =  \  rt/cf)  (11) 

respectively  by  iniinitesimals  of  higher  order,  and 

A=   f  '  !/'h-     and     .1  =  .\    f  "  r-'/cf>  (IV) 

are  therefore  the  formulas  for  the  area  undei'  a  curve  and  between  two 
ordinates,  and  for  the  area  between  the  curve  and  two  radii.  If  the  plant; 
is  ruled  by  lines  parallel  to  lioth  axes  or  l)y  lines  issuing  from  the  pole 
and  by  circles  concentric  with  the  i)ole,  as  is  customary  for  double  inte- 
gration (§§  131,  13-t),  the  increments  of  area  differ  respectively  In' 
iniinitesimals  of  higher  oi'der  from 

dA=<lr,J;/      and       r/.l  =/v//v/c^^  (12) 

and  the  formulas  for  the  area  in  the  two  cases  are 

A  =  lim  2)  A.l  -  fC.I  I  =  fCh'/f/,  (12') 

A  =  lim  V  A.  I  =  fC/.  I  =  rf/v/yv/t^, 

wliere  the  double  integrals  are  extended  over  the  area  desired. 

The  elements  of  volume  whieli  ai'e  reipiireil  foi'  triple  integration 
(-<ij  13,3,  134)  over  a  volume  in  space  may  readily  be  wi'itten  down  i'oi' 
tlie  three  cases  of  rectangular,  polar,  and  cylindrical  cotu'dinatt's.  In  the 
iirst  case  spacer  is  su])})Osed  to  lie  divided  up  ly  ]ilanes  .r  =  i/,  //  =  />, 
z  —  ('  perpendi<'ular  to  the  axes  and  spaced  at  infinitesimal  intervals;  in 
the  second  case,  tlie  division  is  made  liy  the  spheres  /■  —  r/  conet'ntric 
with  the  i)ole.  the  ])lanes  c/)  =  A  through  tlie  ])olar  axis,  and  tlie  cones 
Q  z=  r  of  revolution  al)i)ut  the  })olar  axis;  in  tlie  third  ease  by  tlie  cylin- 
ders r^~<i,  the   planes  ^=^>.  and  the  planes  ::  =  c.    The  infinitesimal 


TAYLOir.S   FORMULA;   ALLIED   TOPICS  81 

volumes  into  Avliicli  space  is  divided  then  differ  from 

(//■  =  (Inh/iJr:,  dr  =  /•-  sin  OdnJifiiW,  dr  =  rdrd(j>dz  (13) 

respectively  l»y  infinitesimals  of  higher  cji'der,  and 


/.rd>/d-:, 


•■-sin  6drd^,ie. 


'  llh 


/cj^d,        (i;V) 


are  the  fornudas  for  the  volumes. 

41.  The  direction  of  a  line  in  space  is  rei)resented  Ijy  the  three  angles 
which  the  line  makes  Avith  the  positive  directions  of  the  axes  or  Ity  the 
cosines  of  those  angles,  tlie  direction  cosines  of  the  line.  From  the  defi- 
nition and  figure  it  ajipears  that 


I  =  cos  a  — 


d.r 

ds 


ill  =  cos  f3 


ds 


n  ^  cos  y  ^ 


dz 
ds 


(14) 


are  the  direction  cosines  of  the  tangent  tn  the  arc  at  the  point;  of  the 
tangent  and  not  of  the  chord  for  the  reason 
that  the  inci'ements  ai'e  ivplaced  by  thediffci-- 
entials.    Hence  it  is  seen  that  foi'  tL('  direc- 
tion cosines  of  the  trnvjcHf  the  propoi'tion 

I  :  111  :  n  =  dx  :  d  ij  :  dz  (14') 

holds.    The  equations  of  a  s})ace  curve  are 

in   terms   of  a    variahle   parameter  /.*    At  the   ]iniiit   (x^^,   >/.,  -;^)  -where 
t=^t_^  the  i'<niiifhins  uf  flir  fiiiKjriit  Jiiirs  ^\()\\\{\  tlirn  1)e 


z 

Axl       / 

p' 

1 

f) 

^Az\y 

Y 

11  —  //,,       ::  —  ^0 


H  —  !/o       ■•  —  -(, 


{''■'%  ('V'.  (''-),       ''      /"'/o)  //'</„)  ^Vo)  ^^'^^ 

As  tlie  cosine  of  the  angle  6  lietween  the  two  directions  given  l)y  the 
direction  cosines  /.  ///,  n  and  I'.  ///.  ;/'  is 

(•()s^= //'  + //////'  +  7/;/,      so      ir  -{-  linn'  +  nn'  =  0  (16) 

is  the  condition  for  the  })ei-pendieularity  of  the  lines,  ^'ow  if  (.'•,  //,  .'.■) 
lies  in  the  plane  normal  to  tlie  curve  at  ;i\^,  //^,,  ,-.■,,.  tlie  lines  determined 
by  tlu'  I'atios  ./•  —  .■;.,  :  //  —  //.,  :  :.'  —  .t;,  and  ("'.'■),-,  :  ("'//)„ :  ('''■-)o  "^^'i^^  ''^'  !"'''■ 
pendicular.    Hence  the  i'<iii(ifti,n  of  tin;  unnnnl.  jdom'  is 

( ,'■  -  ,/• ,) {d..:)^^  +  (.'/-  //„ ) ( 'hl\,  +  ( -  -  -,, ).(//-)o  =  *^> 
or  _/"( t :)(.!'  —  ,/;)  +  y'l  fjii/  —  1/^^ )  +  //'( tj{  ,v  —  zj  =  0.  (17) 

*  For  Tho  suki'  of  L;i'iiiTality  the  piiraiin'trir  foi-m  iii  /  is  assunicil  :  in  a  ]iartiiT,lar  case  a 
simplitiratidii  huliIiT  Ik-  inailf  l>y  taking-  (Hie  of  tlic  vai'iahles  as  f  and  on.'  of  tiic  functicins 
/'',  '/'.  A'  woiiM  tiirn  lit-  1.    TIuis  in  Kx.  S  (e).  //  should  lnj  taken  as  t. 


82 


DI FFE REXTI AL  CxVLCUL US 


The  finKjtnt  i)hine  to  the  curve  is  not  determinate;  an}-  plane  through 
the  tangent  line  will  Ije  tangent  to  the  curve.  If  A  l)e  a  parameter,  the 
pencil  of  tangent  planes  is 


+  A 


0. 


There  is  one  particular  tangent  plane,  called  t/te  osrulfifln'j  plcnt'j^yhuAi 
is  of  especial  importance.    Let 

^  -  -'-o  =  /'(g  r  +  i/"(g  T^  +  i/""(0  r^     r  =  t-  t,      r<$<  ^ 
with   similar  ex})ansions  for  //  and  ,-;,  Ije  the  Taylor  developments  of 
.r,  )/,  z  about  the  point  of  tangency.    When  these  are  substituted  in  the 
equation  of  the  plane,  the  result  is 


1    , 


"/"(g^^r/'vg 


f{Q      y'(^) 


(1+^) 


h"{K) 


-\- 


(1+A) 


i'-"'(0' 


This  expression  is  of  course  proi)orti()nal  to  the  distance  from  any  ])oint 
.7',  )/,  z  of  the  curve  to  the  tangent  })laiie  and  is  seen  to  be  in  general  of 
the  second  order  Avith  respect  to  t  (h-  Ox.  It  is,  however,  ])0ssil)le  to 
choose  for  A  that  value  which  makes  the  iii'st  bracket  vanish.  The  tan- 
gent plane  thus  selected  has  the  [)ro])erty  tliat  fht;  (I'tstifnci'  of  flie  nirrc 
from  if  in  flic  ncifjlihorlKKid  of  flic  iKiinf  of  tinKicnri/  is  of  t/te  fhirddrdcr 
and  is  rnlled  the  osrulatinfj ^lOnn'.    The  suljstitution  of  the  value  of  A  gives 


/"(g   r/'^g    ^''(>„) 
/"(g   f/"(g   ^"(g 


^/•  —  ,'•.        1/  —  1/        Z  —  ,i\  ' 
0       or      \{d.r)J       (d;,)_  (./,V),/'!  =  0       (18) 

'('/^'■),    ('/V)„    ('^^)S 

or  (dn'T-z  -  dzJ'f/)J:r  -  ,,;^)  +  (dzd-.r  -  <l.rd-z)j!/  -  //J 

-^(,l.nP./-d;,d-.r)^IZ-Z^^)  =  0 

as  the  equation  of  the  osculating  plane  In  case /'"(  ^  )  =  .'/"(  z",,)  =  //"(V,,)  =  0, 
this  equation  of  the  osculating  jtlane  vanishes  identically  and  it  is  neces- 
sary to  push  the  development  further  (Ex.  llj. 

42.  For  the  case  of  ]ilane  curves  the  oirnitiiri'  is  dctined  as  the  I'ate 
at  which  the  tangent  turns  com])ared  with  the  description  of  arc,  that 
is,  as  i/(f}/ds  if  d(fi  denotes  the  differential  of  the  angle  througli  which 
the  tangent  turns  when  tlie  ])oint  of  tangency  advances  along  the  cui've 
by  ds.  The  radius  of  curvature  Ji  is  the  reciprocal  of  tlu^  cui-vature, 
that  is.  it  is  ils/d^.    Then 


(/<^  =  ,/tan-' 


ds 


d^  d.r 
dl-^.ls 


^  +  U"J 


R 


'  (I'J) 


TAYLOR'S  FORMULA;  ALLIED  TOPICS 


83 


where  accents  denote  differentiation  with  respect  to  a".  For  space  curves 
the  same  definitions  are  given.  If  /,  m,  n  and  1  -{-dl,  m  -\-diii,  n-{-dn 
are  the  direction  cosines  of  two  successive  tangents, 


But 

Hence 

1   _ 
11-  ~ 


cos  d(^  =  I(J  -\-  dl)  +  III  {ill  -\-  dill)  +  n(7i  -|-  dii). 
P  +  III-  +  u-  =  1     and     (/  +  diy-  +  (in  +  dm)-  +  (ii  +  dn)-  =  1. 
dl-  +  dm-  +  dii-  =  2  —  2  cos  d4>  =  (2  sin  h  dcfif, 


2  sin  \  d(f) 
ds  ~" 


dl-  +  <]iir  +  dir 
d? 


/'-+  m'-  +  n'-,  (19') 


where  accents  denote  differentiation  Avitli  respect  to  s. 

The  torshni  of  a  S})ace  curve  is  defined  as  tlie  rate  of  turning  of  tlie 
osculating  plane  compared  with  the  increase  of  arc  (that  is,  difz/ds,  where 
(/ij;  is  the  ditt'erential  angle  the  normal  to  the  osculating  ]jlane  turns 
through),  and  may  clearly  be  cakadated  ])y  the  same  formula  as  the 
curvature  jjrovided  the  direction  cosines  L,  M,  X  of  the  normal  to  the 
plane  take  the  places  of  the  direction  cosines  /,  in,  n  of  the  tangent  line. 
Hence  the  torsion  is 


lxP\^     dL-  -f-  ilM-  -f  <IX- 


ds 


d.r 


=  Z'-  +  J/'-  +  X'^; 


(20) 


and  the  radius  of  torsion  R  is  defined  as  the  reciprocal  of  the  torsion, 
Avliere  from  the  e(piation  of  the  osculating  plane 


M 


N 


dijd'-':  —  dzd'-i/        dzd'-.r  —  drd'-z 


d.rd'-ij  —  il  ijd'X 
1 


(20') 


V sum  of  S(piares 
The  actual  computation  of  these  quantities  is  somewhat  tedious. 

llie  vectorial  discussion  of  curvature  and  torsion  (§  77)  ^ives  a  better  insight 
into  the  principal  directions  connected  witli  a  space  curve.  These  are  the  direction 
of  the  ((UKicnt.  that  of  the  normal  in  the  oseulatinii'  plane  and  directed  towards 
the  concave  side  of  tlie  curve  and  called  the  principal  normal,  and  that  of  the 
normal  to  the  oseulatin<j,-  plane  drawn  ui)on  that  side  which  makes  the  three  direc- 
tions form  a  right-handed  system  and  called  the  hinoriiud.  In  the  notations  there 
f;'iven.  combined  with  those  aljove. 


r  =  ,ci  +  //i  -h  2k,        t  =  /i  -1-  rn)  +  i<k,        c  =  Xi  +  ^j  -f-  ^^k, 


lA  +  3/j  +  ,Vk, 


when,'  \.  ,u.  "  ure  taken  as  the  direction  cosines  i.if  the  principal  normal.     Now  dt 
is  parallel  h)  c  and  dn  is  i)arallel  to  —  c.    Hence  the  results 


(21) 


dl 

dm 

(7);       '7.S 

dL 

dV 

dX 

r/.s 

—  r^ 

—  -    -    ZZ 

:   :zr  — 

and 

. 

_    — 

-    — 



\ 

M 

"        R 

\ 

fJ- 

•'' 

K 

84 


DIFFEREXTIAL  CALCULUS 


follow  from  dt/ds  =  C  and  dn/ds  =  T.  Now  dc  is  perpendicular  to  c  and  hence  in 
the  plane  of  t  and  n  ;  it  may  be  written  as  dc  =  (t.dc)t+ (n.tZc)n.  Eut  as  t.c  =  n.c  =  0, 
t.dc  =—  C'dt  and  n.(Zc  =—  CfZn.     Hence 

•  (c.(Zt)t  -  (c '7n)n  =  -  Ctds  +  Tnds 


dc 


—  d.s  +  -  (i.s. 

i:         R 


Hence 


(22) 


d\  _       I        L  dix  _      in       ^r  dv  _       n       X 

'ds'~~  Tl^l\'  dl'^Jl^^R'  (Ts  ^  "  i.'  "^  R  ' 

Formulas  (22)  are  known  as  Frenet's  Formuhis;  they  are  usually  written  with  —  R 

in  the  place  of  R  because  a  left-handed  system  of  axes  is  used  and  the  torsidi;.  beini; 

an  odd  function,  changes  its  si^n  when  all  the  axes  are  reversed.    If  accents  dennte 

differentiation  by  .s, 

'         !/'         Z'     \ 


above  fornmlas.  — 


ri'dit-handed 


R 


X        y        z 
x"      y"      z' 

x"--\-y"-  +  z' 


usual  formula^ 
left-handed 


y       z 
1  ,/•'"     //'"     z'" 

-    ;,---,7, — ,,-r  ^) 
£  -  +  u  '-  +  Z  - 


EXERCISES 

1.  Show  that  in  polar  coordinates  in  the  plane,  the  tansent  of  the  inclination 
of  the  curve  to  the  radius  vector  is  rdrp/dr. 

2.  Verify  (10),  (10')  by  direct  transfdnnation  of  corirdinates. 

3.  Fii;  in  the  steps  omitted  in  the  text  in  reuard  i<<  tiu-  pi-oof  nf  (10).  (10')  by 
the  method  of  infinitesimal  analysis. 

^.  A  rhumb  line  on  a  sphere  is  a  line  which  cuts  all  the  meridians  at  a  constant 
angle,  say  a.  Show  that  for  a  rhundj  line  sin  Odfp  —  \-a\\  adO  and  (Zs  =  raecadO. 
Hence  fintl  the  ecpuitidU  (if  the  line,  show  tliat  it  cnils  indehnitely  amund  tlie 
poles  of  the  sphere,  and  that  its  total  length  is  Trr  sec  a. 

5.  Show  that  the  surfaces  represented  liy  F(rf>.  B)  —  0  and  F(r.  0)  ~  0  in  polar 
coordinates  in  space  are  respet'tively  citnes  and  .--urtares  i>f  re\nluti(in  abdut  tlie 
pdlar  axis.    What  sort  of  surface  wuuld  tlie  eijuatinn  F[r.  (p)  =  0  re})resent ".' 

6.  Show  accurately  that  the  expression  -iven  iov  the  differential  of  area  in 
l)olar  coordinates  in  the  plane  and  for  the  diiYerentials  of  \-olunu'  in  polai'  and 
cylindrical  coordinates  in  space  differ  from  the  corre>}iouilin--  increments  by  in- 
linitesimals  of  hiuher  order. 

7.  Show  that  -— ,   r— .   rsin^  —  are  the  direction  cosines  of  the  tauuent  to  a 

(/.S         (/.s-  ds 

space  curve  relative  to  the  radius,  meridian,  and  pai'allel  of  hititude. 

8.  Find  the  tani:'ent  line  and  normal  plane  of  tliese  curves. 

{a)  xyz  =  1.   //-  =  ./•  at   (1.  1,  1).         (/:()  ./•  =  cos  /.   //  =  sin  f.  z  ^  It. 

(y)  2  <nj  =  .(■-.  0  (i-z  =  x'\  (5)  x  =  i  c<  >s  t.   ;/  =  /  sin  /.  ,r  =  Id. 

(e)  y  =  X-.  2'^  =  ]  -  //.  (f)  X-  +  y-  +  z-  =  <(■-.  X-  +  //^  +  2  </./•  =  0. 

9.  Find  the  equation  of  tlie  osculatim:'  jilane  in  the  i'xaiiiiili_'s  of  V.x.  8.  Note 
that  if  X  is  the  independent,  \ariabU'.  the  equation  of  the  jihine  is 

/dy  d-z  _  dz  d-y\ 
\dx  dx-      dx  dx- 


(-  -  ^u)  =  0- 


TAYLOR'S  FORMULA;  ALLIED  TOPICS  85 

10.  A  space  curve  passes  throuich  the  origin,  is  tangent  to  the  j;-axis,  and  has 
z  =  0  as  its  (jsculating  phine  at  tlie  origin.    Siiow  tliat 

■I  =  tr{0)  +  \  f\r{0)  +  •  ■  • ,         y^i  t^g"{0)  +  ■■■  ,         z  =  1  t^h'"{0)  +  •  •  • 

will  be  the  form  of  its  Maclaurin  development  if  ^  =  0  gives  x  =  y  =^  z  =  0. 

IL  If  the  2d.  3d,  •  •  ■ ,  (u  —  l)st  derivatives  of  /,  g,  h  vanish  for  t  =  t^  but  not 
all  the  ?ith  derivatives  vanish,  show  that  there  is  a  plane  from  which  the  curve 
departs  by  an  infinitesimal  of  the  {n  +  l)st  order  and  with  which  it  therefore 
has  contact  of  order  n.  Such  a  plane  is  called  a  hyperosculating  plane.  Find  its 
eijuation. 

12.  At  what  pr)ints  if  any  do  the  curves  (/3),  (7),  (e),  (f),  Ex.  8  have  hyperoscu- 
lating planes  and  what  is  the  degree  of  contact  in  each  case  ? 

13.  Show  that  the  expression  for  the  radius  of  curvature  is 

--  \  X   -  +  1/   -  +  ~ 3 , 

A  If'^  +  y'^  +  k'-]^ 

where  in  the  iirst  case  accents  denote  differentiation  by  .s,  in  the  second  by  t. 

14.  Show  that  the  radius  of  curvature  of  a  space  curve  is  the  radius  of  curva- 
ture of  its  projection  on  the  osculating  plane  at  the  point  in  (piestion. 

15.  Fr(jni  Frenefs  Formulas  show  that  the  successive  derivatives  ol'  x  are 

x'  =  /,         x"  = 


X 

,       X' 

\ir 

I 

>  /'          /. 

—  , 

x'" 

—  — 

— 



-  X~  + 

It 

R 

1!^ 

A'-i 

R^     i:r 

where  accents  denote  differentiation  by  .s.  Slmw  that  the  results  fru-  y  and  z  are 
the  same  except  that  //(.  ix.  M  or  n.  v.  X  take  the  places  of  I.  X.  L.  Hence  infer 
that  fur  tlie  xth  derivatives  the  results  are 

j:(")  =  IJ\  +  XP^  -f-  LP...        !/<")  -=  inl\  +  nP..  4-  MP„.        z(">  =  nP^  +  vP.^  -f  XP,., 

where  /'j.  /■".,.  /*.  are  rati(jnal  funt.-tioiis  of  U  and  R  and  their  derivati\es  by  s. 

16.  Apply  the  foregoing  to  the  expansion  of  Kx.  10  to  sliow  that 


]     .. 

s- 

ir   „ 

-r^  ■'*■'  +  •••' 

y  = 



--7-,.^-  + 

0  !i^ 

2  1! 

(i  It- 

G  y^'R 

where  A'  and  R  are  the  values  at  the  origin  where  .s  =  0,  I  =  ^  =  X  =  1,  and  the 
other  six  direction  cosines  m.  n.  X.  v.  L.  M  vanish.  Find  .s  and  write  the  expan- 
sion of  the  curve  of  Ex.  8  (7)  in  this  form. 

17.   Note  that  the  distance  of  a  point  on  the  curve  as  expanded  in  Ex.  10  from 
the  sphere  through  the  origin  and  with  center  at  the  point  (0,  It.  It'R)  is 


Vx-  +  {!/-  It)-  +  {z  -  It'Rf  -  V/.'-  +  /,"-R- 

(.r-  -I-  if^  -  -1  Uy  -i^  z-  -2  Tt'Rz) 


V  .f^  +  {y-  Itf  +{z-  It'Rf  ^  \  R  +  /."-R-^ 

and  consequently  is  of  the  fc.iurth  (jrder.  The  curve  therefore  has  contact  of  the 
third  order  with  this  sphere.  Can  the  e(jUation  of  this  sphere  be  derived  by  a 
limiting  process  like  that  of  V.x.  18  as  a^jplied  to  the  osculating  plane 


80 


DIFFERENTIAL  CALCULUS 


18.  The  osculatinj,'  plane  may  be  regarded  as  the  plane  passed  through  three 
consecutive  points  of  the  curve  ;  in  fact  it  is  easily  shown  that 


lini 

&r,  6?/,  Sz 
Ax.  A?/,  Az 
approach  0 


X  y  Z  I 

•^0  Vo  ^0  1 

■r,,  +  Sx  ?/,j  +  dy  Zq  +  5z  1 

/y  +  Ax  l/o  +  A;/  z^^  +  Az  1 


^-^0  y-Vo  z-^o 

(dx)o        (d2/)o        (dz)^ 
{(Px)^      (d^y)^      {dh)^ 


=  0. 


19.  Express  the  radius  of  torsion  in  terms  of  the  derivatives  of  x,  y,  z  hy  t 
(Ex.  10,  p.  07). 

20.  Find  the  direction,  curvature,  osculating  plane,  torsion,  and  osculating 
sphere  (Ex.  17)  of  the  conical  helix  x  =  t  cos  t.  y  =  t  sin  t,  z  =  kt  at  i  =  2  tt. 

21.  Upon  a  plane  diagram  which  shows  A.s,  Ax.  Ay,  exhibit  the  lines  which 
represent  ds,  dx,  dy  under  the  different  hypotheses  that  x,  y,  or  s  is  the  independ- 
ent variable. 


CHAPTER  IV 
PARTIAL  DIFFERENTIATION;   EXPLICIT  FUNCTIONS 

43.  Functions  of  two  or  more  variables.  The  definitions  and  theo- 
rems about  functions  of  more  than  one  independent  variable  are  to  a 
large  extent  similar  to  those  given  in  Chap.  II  for  functions  of  a  single 
variable,  and  the  changes  and  difficulties  which  occur  are  for  tlu^  most 
part  amply  illustrated  by  the  case  of  two  varia])les.  The  work  in  the 
text  Avill  therefore  be  confined  largely  to  this  case  and  the  generaliza- 
tions to  functions  involving  more  than  two  variables  may  be  left  as 
exercises. 

If  the  value  of  a  variable  z  is  uniquely  determined  when  the  values 
(.'-,  y)  of  two  variables  are  known,  ,';  is  said  to  be  a  function  z  =f(^.r,  y) 
of  the  two  variables.  The  set  of  values  [(,r,  y)]  or  of  points  P{.r,  //)  of 
the  .'/'y-plane  for  which  z  is  defined  may  be  any  set,  but  usually  consists 
of  all  the  points  in  a  certain  area  or  region  of  the  i)lane  bounded  by 
a  curve  which  may  or  may  not  belong  to  the  region,  just  as  the  end 
points  of  an  interval  may  or  may  not  belong  to  it.  Thus  the  function 
1/ V  1  —  y^  —  if  is  defined  for  all  points  within  the  circle  .'•"  -f-  //"  =  1, 
but  not  for  points  on  the  perimeter  of  the  circle.  For  most  ])urposes  it 
is  sufficient  to  think  of  the  boundary  of  the  region  of  dt'hnition  as  a 
polygon  whose  sides  are  straight  lines  or  such  curves  as  the  geometric; 
intuition  naturally  suggests. 

The  first  way  of  representing  the  function  z  =/(./•,  y)  geometrically 
is  by  the  Hiirfdce  z  =f(;t,  //),  just  as  //  =t\.r'^  was  represented  by  a  curve. 
This  method  is  not  available  for  u  =/(./•,  //,  z),  a  function  of  three  vari- 
ables, or  for  functions  of  a  gi-eater  numlx^r  of  variaV)les  ;  for  space  has 
only  three  dimensions.  A  second  method  of  representing  the  function 
z  =/{.>■,  ij)  is  by  its  contmir  lines  in  the  ./-y-plane,  that  is,  the  curves 
/{■''■:  !/)  =  const,  are  plotted  and  to  each  curve  is  attached  the  value  of 
the  constant.  This  is  the  method  employed  on  maps  in  marking  heights 
above  sea  level  or  depths  of  the  ocean  below  sea  level.  It  is  evident  that 
these  contour  lines  are  nothing  but  the  projections  on  the  .rv/-plane 
of  the  curves  in  which  the  surface  z  =f{x,  y)  is  cut  by  the  planes 
z  =  const.  This  method  is  a})})licable  to  functions  u  =  f(.r,  //,  z)  of 
three  variables.    The  contour  surf  (ires  u  =  const,  whic-h  are  thus  obtained 

87 


DIFFEKEXTIAL  CALCULUS 


are  frequently  called  c'luqioteiituil  Hurfact^s.     If  the  function  is  single 
valued,  the  contour  lines  or  surfaces  cannot  intersect  one  another. 

The  function  z  =f(.r,  y)  is  continuous  for  (u,  h)  Avhen  either  of  the 
following  equivalent  conditions  is  satisfied : 

1°.        lini /'(•'•,  U)  =/("!  ^')     ^"'     li"\/'(''^j  .'/)  =  ,f  G"'^^  •'■'  1"'^  2/)j 
no  iiKittcr  lioir  tin;  ra ru Oil c  point  i^(.'',  y)  (tpproitchcs  (ji,  //j. 
2°.   If  for  any  assigned  e,  a  niiuihcr  8  may  Jx'  found  so  tltat 

!/(■'•, ii) - fd',  W\<^    "'^' '-'''    I •'•  - "  1  < s.  i y - ^' ; < s- 

Geometric-ally  this  means  that  if  a  square  with  {a,  h)  as  center  and 


/(a,WH 


25 


\       -^ 
/{a,W-e 


with  sides  of  length  2  8  parallel  to  the  axes  be  drawn, 

the   portion   of  the   surface  z  =  f{.r,  y)   aljove   the 

square  will  lie  Ijetween  the  two  planes  z^f(a.  l/)±e. 

Or  if  contour  lines  are  used,  no  line  f(.r,  y)  =  const. 

where  the  constant  differs  from /'('/,  h)  1a'  so  much 

as  e  Avill  cut  into  the  square.    It  is  clear  that  in  place     0\  25       X 

of  a  square  surrounding  ((/,  />)  a  circle  of  radius  8  or  any  other  figure 

wdiicli  lay  within  the  square  might  be  used. 

44.  Contlnultij  examinc-'I.  From  the  deliiiition  of  continuity  just  <iivon  and 
from  tlie  corresponding  definition  in  §  24,  it  follows  that  if  /(x,  y)  is  a  continuous 
fnnction  of  x  and  y  for  (a,  h),  then /(j,  h)  is  a  continuous  function  of  x  for  x  =  a 
and  f(a.  y)  is  a  contiinu.ius  function  of  y  for  y  =  h.  Tluit  is.  if  /  is  continuous  in 
x  and  y  jointh',  it  is  contiiuKjus  in  x  and  y  severally.  It  mii^lit  be  thought  that 
conversely  if  f{x.  h)  is  contiiuious  for  x  =  a  and  f{a.  y)  for  y  =  h.  fix.  y)  would 
be  continuous  in  (/.  y)  for  (u.  h).  Tliat  is,  if  /  is  continuous  in  x  and  y  severally, 
it  would  be  continuous  in  x  and  y 
jointly.  A  simple  example  will  show 
that  this  is  not  necessarily  true.  Con- 
sider the  case 


z  =f(x,  y)  = 


X-  +  y- 


•1^  +  y 
/(O.  0)  =  0 

and  examine  z  for  coutiiuiity  at 
(0.  0).  I'he  functions  f(x.  0)  =  .r. 
anil/(0,  y)  ~  y  are  surely  continuous 

in  their  respective  variables.  But  the  surface  z  =  fix.  y)  is  a  conical  surface  (except 
for  the  points  of  the  z-axis  other  than  the  origin)  and  it  is  clear  that  Fix.  y)  may 
approach  the  origin  in  such  a  manner  that  z  shall  approach  any  di'>ired  value. 
^Moreover,  a  glance  at  the  comour  lines  shows  that  they  all  enter  any  circle  or 
square,  no  matter  how  small,  concentric  with  the  origin.  If  P  approaches  the  origin 
along  one  of  these  lines,  z  remains  constant  and  its  limiting  value  is  tliat  constant. 
In  fact  by  approaching  the  oriuiu  alon^  a  set  of  p((ints  which  juiup  from  one  con- 
tour line  to  another,  a  method  of  aiijiroach  luay  be  found  surli  that  z  ajiproaches 
no  limit  whatsoever  but  oscillates  between  wide  limits  or  becomes  iulinite.  Clearly 
the  conditions  (,f  contimiity  are  not  at  all  fultilled  by  z  at  (0,  0). 


PARTIAL  DIFFEREXTIATIOX ;   EXPLICIT 


89 


Double  limits.   There  often  arise  for  consideration  expressions  like 


lini 


riini/(x,  2/)l.         lini  riini/{j,  y)!, 


(1) 


■where  the  limits  exist  whether  x  first  approaclies  its  limit,  and  then  y  its  limit,  or 
vice  versa,  and  where  the  question  arises  as  to  whether  the  two  limits  thus  obtained 
are  equal,  that  is,  whether  the  order  of  taking  the  limits  in  the  double  limit  inay 
be  interchanged.  It  is  clear  that  if  the  function /(x,  y)  is  continuous  at  {a.  h),  the 
limits  approached  by  the  two  expressions  will  be  ecjual ;  for  the  limit  of  /(.r,  y)  is 
/(«,  b)  no  matter  how  (j,  y)  approaches  (a,  h).  If  /  is  discontinuous  at  («,  b),  it 
may  still  happen  that  the  order  of  the  limits  in  the  double  limit  may  be  inter- 
changed, as  was  true  in  the  case  above  where  the  value  in  either  order  was  zero  ; 
but  this  cannot  be  afhi'med  in  general,  and  special  c(jnsiderati(jns  must  be  applied 
to  each  case  when /is  discontinuous. 

Varieties  of  reyioriH*  For  both  pure  mathematics  and  physics  the  classilication 
of  regions  according  to  their  connectivity  is  important.  Consider  a  finite  region  li 
bounded  by  a  curve  which  nowhere  cuts  itself.  (For  the  present 
pui-poses  it  is  not  necessary  to  enter  upon  the  subtleties  of  tlie 
meaning  of  "curve"  (see  §§  127-128);  ordinary  intuition  will 
suffice.)  It  is  clear  that  if  any  closed  curve  drawn  in  this  region 
had  an  unlimited  tendency  to  contract,  it  could  draw  togethei- 
to  a  point  and  disappear.  On  the  other  hand,  if  1'/  be  a  region 
like  E  except  that  a  portion  has  been  removed  so  that  W  is 
bounded  by  two  curves  one  within  the  other,  it  is  clear  tluit 
some  closed  curves,  namely  tliose  which  did  not  encircle  the 
portion  removed,  could  shrink  away  to  a  point,  whereas  (jther 
closed  curves,  namely  those  which  encircled  that  portion,  could 
at  most  shrink  down  into  coincidence  with  the  boundary  of  that 
portion.  Again,  if  two  portions  are  removed  so  as  to  give  rise 
to  the  region  E'\  there  are  circuits  around  each  of  the  p(jrtions 
which  at  most  can  only  shrink  down  to  the  boundaries  of  those 
portions  and  circuits  around  both  portiims  which  can  shrink  down  to  the  bounda- 
ries and  a  line  joining  them.  A  region  like  A',  where  any  closeil  curve  or  circuit 
may  be  shrunk  away  to  nothing  is  called  a  simply  connected  reybm  ;  whereas  regions 
in  which  there  are  circuits  whicli  cannot  be  shrunk  away  to  nothing  are  called 
inulVqAy  connected  regions. 

A  multiply  connected  region  may  be  made  simply  connected  \)X  a  simple  device 
and  convention.  For  .suppose  that  in  IV  a  line  were  drawn  connecting  the  two 
bounding  curves  and  it  were  agreed  that  no  curve  or  circuit  drawn  within  /."  should 
cross  this  line.  Then  the  entire  region  woidd  be  surnnuided  by  a 
single  boundary,  part  of  whicli  would  be  C(junted  twice.  The  figure 
indicates  the  situation.  I]i  like  manner  if  two  lines  were  drawn  in 
It"  connecting  both  interirtr  Ijoundarles  to  the  exterior  or  connecting 
the  two  interi(jr  Ixjundaries  together  and  eitlier  of  them  to  tlie  outer 
boundary,  the  region  would  be  rendered  simply  connected.  The  entire  region 
would  have  a  single  boundary  oi  whicli  x»arts  would  be  coiuited  twice,  and  an\' 
circuit  which  did  iKjt  cross  the  lines  could  be  shrunk  awav  to  nothinu.    The  lines 


*  The  discussion  from  this  pcjint  to  the  end  of  §  45  may  be  connected  witli  that  of 
§§  123-126. 


90 


DIFFElt EX TI AL  CALCULUS 


thus  drawn  in  the  region  to  make  it  simply  connected  are  called  cuts.  There  is  no 
need  that  the  region  be  Unite  ;  it  miglit  extend  off  indetinitely  in  some  directions 
like  the  region  between  two  parallel  lines  or  between  the  sides  of  an  angle,  or  like 
the  entire  half  of  the  jy-plane  for  which  y  is  positive.  In  such  cases  the  cuts  may 
be  drawn  either  to  the  boundary  or  off  indefinitely  in  such  a  way  as  not  to  meet 
the  boundary. 

45.  Multiple  calued  fiDictions.  If  more  than  one  value  of  z  corresponds  to  the 
pair  of  values  (/,  y),  the  function  z  is  multiple  valued,  and  there  are  some  note- 
worthy differences  between  nuiltiple  valued  functions  of  one  variable  and  of  several 
variables.  It  was  stated  (§  23)  that  multiple 
valued  functions  were  divided  into  branches 
each  of  which  was  single  valued.  There  are 
two  cases  to  consider  when  there  is  one  vari- 
able, and  they  are  illu.strated  in  the  figure. 
Either  there  is  no  valtie  of  x  in  the  interval 
for  which  the  dift'erent  values  of  the  function 
are  equal  and  there  is  consequently  a  number 
D  which  gives  the  least  value  of  the  difference 
between  any  two  branches,  or  there  is  a  value  of  x  for  which  different  branches 
have  the  same  value.  Now  in  the  first  case,  if  x  changes  its  value  continuously  and 
if/(j:)  be  constrained  also  to  change  contiimously,  there  is  no  possibility  of  passing 
from  one  branch  of  the  function  to  another  :  but  in  the  second  case  such  change  is 
possible  for,  when  x  pas.ses  through  the  value  for  which  the  branches  have  the  same 
value,  the  function  while  constrained  to  change  its  value  contiimously  may  turn  off 
onto  the  other  branch,  althottgh  it  need  not  do  so. 

In  the  case  of  a  function  z  =f{x.  y)  of  two  variables,  it  is  not  true  that  if  the 
values  of  the  function  nowhere  become  equal  in  or  on  the  boundary  of  the  region 
over  which  the  function  is  defined,  then  it  is  impossible  to  pass  continuousb'  from 
one  branch  to  another,  and  if  P{x.  y)  describes  any 
continuous  closed  curve  or  circuit  in  the  region,  the 
value  of  f{x.  ij)  clianging  contiiniously  must  return  to 
its  original  value  when  P  has  completed  the  descrip- 
tion of  the  circuit.  For  suppose  the  function  z  be  a 
lielicoidal  surface  z  =  a  tan-i(/y/,f).  or  rather  the  por- 
tion of  tliat  surface  between  two  cylindrical  surfaces 
concentric  with  the  axis  of  the  heliroid.  as  is  the  case 
of  the  surface  (if  the  screw  of  a  jack,  and  the  circuit 
be  taken  around  the  inner  cylinder.  The  nmltiple  num- 
bering of  tlie  contour  lines  indicates  the  fact  that  the 
function  is  mulriple  valued.  Clearly,  each  time  that 
the  circuit  is  ilescriljed.  the  value  of  z  is  increased  liy  the  amount  between  the  >uc- 
cessive  branches  or  leaves  of  the  surface  (or  decreased  by  that  amount  if  the  cinMiit 
is  descriljed  in  the  opposite  direction).  The  region  here  dealt  with  is  not  simply 
connected  and  the  circuit  cannot  be  shrunk  to  nothing  —  which  is  tlie  key  to  the 
situatif.in. 

Thkohkm.  If  the  difference  between  the  different  values  of  a  continuous  mul- 
tiple valued  function  is  never  less  than  a  finite  number  1)  for  any  set  (./•.  y)  of 
values  of  the  variables  whetlier  in  or  upon  the  boundary  of  the  rcirioii  of  detini- 
tion.  then  the  value  f{x.  y)  of  the  fiuiction.  constrained  to  change  continuously, 


70,27r 


PARTIAL  DIFFEEEXTIATIOX;  EXPLICIT 


91 


will  return  to  its  initial  value  when  the  point  P{x.  ij).  (lescribinu-  a  closed  curve 
which  can  be  shrunk  to  nothing,  completes  the  circuit  and  returns  to  its  starting 
point. 

Now  owing  to  the  continuity  of  /  throughout  the  region,  it  is  possible  to  find  a 
numbers  so  that  \f(x,y)  —  f{x'.y')\<e  when  |x  — J'']<5  and  |*/—  ?/'|<5  no  matter 
what  points  of  the  region  (x,  y)  and  (x',  y')  may  be.  Hence  the  values  of  /  at  any 
two  points  of  a  small  region  which  lies  within  anj'  circle  of  radius  \  5  cannot  differ 
by  so  nuich  as  the  amount  !>.  If,  then,  the  circuit  is  so  small 
that  it  may  be  inclosed  within  such  a  circle,  there  is  no  possi- 
bility of  passing  from  one  value  of  /  to  another  when  the  circuit 
is  described  and  /  nmst  return  to  its  initial  value.  Next  let 
there  be  given  any  circuit  such  that  the  value  of /.starting  from 
a  given  value /(x,  y)  returns  to  that  value  when  the  circuit  has 
been  completely  described.  Suppose  that  a  modification  were 
introduced  in  the  circuit  by  enlarging  fir  diminishing  the  inclosed  area  l)y  a  sniaii 
area  lying  wholly  within  a  circle  of  radius  I  8.  Consider  the  circuit  ABCDEA  and 
the  modified  circuit  ABCDEA.  As  these  circuits  coincide  except  for  the  arcs  B(  '1) 
and  BCD.  it  is  only  necessary  to  show  that/  takes  on  tlie  same  value  at  D  whether 
D  is  reached  from  B  by  the  way  of  C  or  by  the  way  of  C .  Jiut  this  is  necessarily 
so  for  the  reason  that  both  arcs  are  within  a  circle  of  radius  J  5. 
Then  the  value  of  /  must  still  return  to  its  initial  value  /(x,  y) 
when  the  modified  circuit  is  described.  Now  to  comi)lete  the 
proof  of  the  theorem,  it  suffices  to  note  that  any  circuit  which 
can  be  shrunk  to  nothing  can  be  made  up  by  piecing  together  a 
number  of  small  circuits  as  shown  in  the  figure.  Then  as  the 
change  in /around  any  one  of  the  small  circuits  is  zerf).  the  change  nnist  be  zero 
around  2,  3,  ■!,•••  adjacent  circuits,  and  thus  finalh'  around  the  complete  large 
circuit. 

Kedurlbllity  of  circuits.  If  a  circuit  can  be  shrunk  away  to  nothing,  it  is  said  to 
be  reducible ;  if  it  camiot,  it  is  .said  to  l)e  irreducible.  In  a  simi^l}-  connected  region 
all  circuits  are  reducible  ;  in  a  multiply  connected  region  there  are  an  infinity  of 
irreducible  circuits.  Two  circuits  are  said  to  be  e(juiv(ile}it  or  I'edvudble  to  each 
other  when  either  can  l)e  expanded  or  shrunk  into  the  other.  The  change  in  the 
value  of /on  passing  antund  two  equivalent  circuits  from  .1  to  .1 
is  the  same,  provided  the  circuits  are  described  in  the  same  direc- 
tion. For  consider  the  figure  and  the  equivalent  circuits  ^ICA 
and  AC  A  described  as  indicated  by  the  large  arrows.  It  is  clear 
that  either  may  be  modified  little  by  little,  as  indicated  in  the 
])r(iof  above,  until  it  has  been  changed  into  the  other.  Hence  the 
cliange  in  the  value  of /around  the  two  circuits  is  the  same.  Or,  as  another  pi'oof, 
it  may  be  observeil  that  the  condjined  circuit  AC\l("A.  where  the  second  is 
described  as  indicated  Viy  the  small  arrows,  may  be  regarded  as  a  reduciljle  circuit 
wliicli  toviches  itself  at  ^l.  Then  the  change  of  /  around  the  circuit  is  zero  and  / 
must  lose  as  much  on  passing  from  A  to  A  by  C  as  it  gains  in  iiassing  from  .1  to 
^1  by  C.  Hence  on  passing  from  .1  to  -1  by  C  in  the  direction  of  the  large  arrows 
the  gain  in/mu.st  be  the  same  as  on  passing  by  C. 

It  is  now  possible  to  see  that  any  circuit  ABC  may  he  reduced  to  rirruitH  uround 
the  portions  cut  out  of  the  region  combined  with  lines  goinij  to  mid  fruin  A  nud  the 
boundaries.    The  figure  shows  this;    for  the  circuit  AB("BAlJ("'DA  is  clearly 


92 


I )  I F  FE  K  EX  T I A  L  ( '  AL(  J  U  LU  S 


rodiu'ible  to  the  circuit  AC  A.  It  must  not  be  forgotten  that  althnuuh  the  lines  yl7i 
and  BA  coincide,  tlie  values  of  the  function  are  not  necessarily  the  same  on  AB 
as  on  BA  but  differ  by  the  amount  of  change  introduced  in 
/on  passing  around  the  irreducible  circnit  BC'B.  One  of  the 
cases  which  arises  most  frequentlj'  in  practice  is  that  in 
which  the  successive  branches  of  /(x,  y)  differ  by  a  constant 
amount  as  in  the  case  z  —  tan-i(///x)  where  2  tt  is  the  differ- 
ence between  successive  values  of  z  for  the  same  values  of  the 

variables.  If  now  a  circuit  such  as  ^IBC'BA  lie  considere<l,  where  it  is  imagined 
that  the  origin  lies  within  BC'B,  it  is  clear  that  the  values  of  z  along  ,17>  and 
along  J}A  differ  by  2  7r,  and  whatever  z  gains  on  passing  from  A  to 
B  will  be  lost  on  passing  from  B  to  ,1,  although  the  values  through 
which  z  changes  will  be  different  in  the  two  cases  by  the  amount  // 
2  7r.  Hence  the  circuit  ABC'B^l  gives  the  same  changes  for  z  as 
the  simpler  circuit  BC'B.  In  other  words  the  result  is  obtained 
that  if  the  different  vdlues  of  a  multiple  vidued  fundinn  for  the  same 
V(dites  of  the  i^arioliles  differ  hy  a  eonstant  independent  of  the  nilues  of 
t/ie  variables,  any  circuit  may  he  reduced  to  circuits  about  the  bound- 
aries of  the  p)ortions  removed;  in  tliis  case  the  lines  going  from  the  point  A  to  the 
boundaries  and  back  mav  be  discarded. 


(a).  =  ;^-  +  |,     .(0,0,.0. 


iii) 


EXERCISES 

1.  Draw  the  contriur  lines  and  sketch  the  surfaces  corresponding  to 

■I'll 
--    -■  — ,     2(0.  0)  =  0. 

Note  that  here  and  in  the  text  only  one  of  the  contour  lines  passes  through  the 
origin  although  au  iutiiiite  luunber  have  it  as  a  frontier  point  between  two  parts 
of  the  same  contour  line.    Discuss  the  doulile  limits  lim  lim  z.  lim  lim  z. 

.r  =  n   II  =  0  //  =  0    .'•  =  0 

2.  Draw  the  contour  lines  and  sketch  the  surfaces  corresponding  to 


(a)  z 


./•-  +  y-  -  1 


(/3) 


(7)  z 


X-  +  2  y-  -  1 


•2y  "  '  X  " '  2  X'-  +  y-  —  1 

Examine  particularly  the  behavior  of   the  function   in  the  neighlxirhood  of  the 
apparent  points  of  intersection  of  different  contour  lines.    Why  apparent'.' 

3.  State  and  prove  for  functions  of  two  independent  variables  the  generaliza- 
tions of  Theorems  (1-11  of  ( 'haii.  1 1.  Note  that  the  theorem  on  luiiformity  is  proved 
for  two  variables  by  the  ap[)licatioii  of  Kx.  0.  p.  40.  in  almost  the  identical  manner 
as  for  the  case  f>f  one  variabh.'. 

4.  ( lutline  definitions  and  theorems  for  functions  of  three  variables.  In  partic- 
ular indicate  the  contour  surfaces  of  the  functions 


(<»)   »  = 


.r  +  y  +  2  z 


(/3)«=4^-^^'  (7)»  =  ^. 


y  -  z  ■  '  X  +  (/  +  2 

and  discuss  the  triple  limits  as  x.  //.  z  in  different  orders  approach  the  origin. 

5.  Let  2  =  V{s.  ]i)/(^{.r.  y).  where  7'  and  fj  ai'i-  ]iolynomials.  lie  a  rational  func- 
tion of  X  and  //.  Show  that  if  the  curves  /'  =  0  and  (^>  ^  0  intersect  in  any  points, 
all  the  contour  lines  of  z  will  ctm verge  toward  these  points  ;  and  conversely  show 


PARTIAL  DIFFEKEXTIATIOX;   EXPLICIT  93 

that  if  twfi  (lifftreiit  contour  lines  of  z  apparently  cut  in  some  point,  all  the  contour 
lines  will  converge  toward  that  point,  P  and  Q  will  there  vanish,  and  z  will  \)c 
undefined. 

6.  If  1)  is  the  minimum  difference  between  different  values  of  a  nndtiple  valued 
function,  as  in  the  text,  and  if  the  function  returns  to  its  initial  value  plus  I)'^D 
when  P  describes  a  circuit,  show  that  it  ■will  return  to  its  initial  value  plus  I)'^D 
when  P  describes  the  new  circuit  formed  by  piecing  on  to  the  given  circuit  a  small 
region  which  lies  within  a  circle  of  radius  J  5. 

7.  Study  the  function  z  =  tan- 1(2///).  noting  especially  the  relation  between 
contour  lines  and  the  surface.  To  eliminate  the  origin  at  which  the  function  is  not 
defined  draw  a  small  circle  about  the  point  (0.  0)  and  observe  tiiat  the  region  of 
the  whole  ^iz-plane  outside  this  circle  is  not  simph'  connected  but  may  be  made  so 
by  drawing  a  cut  from  the  circumference  off  to  an  infinite  distance.  Study  the 
variation  of  the  function  as  P  describes  various  circuits. 

8.  Study  the  contour  lines  and  the  surfaces  due  to  the  functions 

(rt)  z  =  tan-ij-y,         ^^^  2  =  tan- ' —,         (7)  ^  =  sin- ^  (./•  —  y). 

1  -  .'/'- 

Cut  out  the  XJoints  where  the  functions  are  not  defined  and  fi)llow  the  chanu'es  in 

the  functions  about  such  circuits  as  indicated  in  the  figures  of  the  text.    IIow  may 

the  region  of  definition  be  made  simply  connected  ? 

9.  Consider  the  function  z  ~  tan-  ^  (P/  Q)  where  P  and  (J  are  polynomials  and 
where  the  curves  P  =  0  and  Q  =  0  intersect  in  11  pnints  (11^.  h^).  {u.,.  h.,).  ■  ■  ■.  (((„.  b„) 
V)ut  are  not  tangent  (the  polynomials  have  cnmnKm  sulutidus  which  are  Udt  mul- 
tiple roots).  Show  that  the  value  of  the  function  will  change  b\'  2  kir  if  (,r,  y) 
describes  a  circuit  which  includes  k  of  the  points.  Illustrate  liy  taking  for  P/Q 
the  fractions  in  Ex.  2. 

10.  Consider  regions  or  volumes  in  space.  Slmw  that  there  are  regions  in  which 
some  circuits  cannot  lie  shrunk  away  t<i  nothing  ;  also  regions  in  which  all  cii'cuits 
may  be  shrunk  away  but  not  all  chised  surfaces. 

46.  First  partial  derivatives.  Let  ,^■  =/'(.'•, //)  1h'  a  single  valuod 
function,  or  one  ],)ninc]i  of  a  multiple  valued  function,  defined  for  (",  Ji) 
and  for  all  points  in  tlie  neigliliorliood.  If  //  be  given  the  value  />, 
then  z  l)ecomes  a  function /'(.'■.  I')  of  .r  alone,  and  if  that  function  has  a 
derivative  for  .'•  =  <i,  that  derivative  is  called  tlie  pdrfhil  dcrii-nflre  of 
,-;;  =/(:/•,  7/)  with  respect  to  ,'•  at  ('/,  //).  Similaily,  if  ,/■  is  held  fast  and 
eiptal  to  a  and  ii /(a.  //)  has  a  deiivative  when  //  =  //.  that  deiivative  is 
called  the  partial  derivative  of  ,v  with  respe<'t  to  >/  at  ('",  //).  To  obtain 
these  derivatives  formally  in  the  case  of  a  given  function  /(,'■,  >/)  it  is 
merely  necessary  to  differentiate  the  function  by  the  ordinary  rtdes, 
treating  //  as  a  constant  Avlien  finding  tht^  derivative  with  respect  to  ./■ 
and  a;  as  a  constant  for  the  dorivativc  with  rcspei't  to  //.    Xotations  are 

T^  =  —  =./;  =/ .  =  <■  =  l\  /  =  i'r-  =    -T 
ex        ex  \(IX 


94  DIFFER  EX  TI A  L  CALCULUS 

for  the  .r-derivative  with  similar  ones  for  the  ^-derivative.  The  partial 
derivatives  are  the  limits  of  the  quotients 

\\m- — ,        lim ,  {1) 

provided  those  limits  exist.  The  application  of  the  Theorem  of  the 
Mean  to  the  functions  f(.r,  />')  and  f(".  //)  gives 

/("  +  /',  /')  -/(",  ^')  =  /'./;:("  +  oj'-  />),  0  <  ^^  <  1, 

f(",  h  + 1-)  -f(",  f>)  =  J^f;Of,  h  +  ej:),  0  <  ^^  <  1,       ^"^ 

under  the  proper  but  evident  restrictions  (see  §  26). 

Two  coinments  niaj'  be  made.  First,  some  writers  denote  the  partial  derivatives 
by  the  same  symbols  dz/dx  and  dz/dy  as  if  z  were  a  function  of  only  one  variable 
and  were  differentiated  with  respect  to  that  variable  ;  and  if  they  desire  especially 
to  call  attention  to  the  other  variables  which  are  held  constant,  they  athx  them  as 
.subscripts  as  shown  in  the  last  symbol  ,i;iven  (p.  D.S).  This  notation  is  particularly 
prevalent  in  thermodynamics.  As  a  matter  of  fact,  it  would  probably  be  impos- 
.sible  to  devi.se  a  simple  notation  for  partial  derivatives  which  should  l)e  satisfac- 
tory for  all  purposes.  The  only  safe  rule  to  adopt  is  to  use  a  notation  which  is 
sutticieutly  explicit  for  the  purposes  in  hand,  and  at  all  times  to  pay  careful  atten- 
tion to  what  the  derivative  actually  means  in  each  case.  Second,  it  should  l)e  noted 
that  for  points  on  the  boundary  of  the  region  of  definition  nf /(.;•.  //)  there  may  be 
merely  right-hand  or  left-hand  partial  derivatives  or  perhaps  none  at  all.  For  it 
is  necessary  that  the  lines  y  =  h  and  x  =  «  cut  into  the  region  on  one  side  or  the 
f)ther  in  the  neighborhood  of  [a.  h)  if  there  is  to  be  a  derivative  even  one-sided  ; 
and  at  a  corner  of  the  boundarj'  it  may  happen  that  neither  of  these  lines  cuts 
into  the  region. 

Theorem.  If  _/'(,'•,  //)  and  its  derivatives  f^  and  f'y  are  continuous  func- 
tions of  (./•,  //)  in  the  neighborhood  of  (ji,  b),  the  increment  A/ may  l»e 
written  in  any  of  the  three  forms 

=  ^'fX"  +  oj'-  h  +  /.:/;;("  +  /'.  /'  +  oj.-) 

=  iif'A"  +  Oh.  h  +  Ok)  +  hr:,("  +  eii.  h  +  Bh)  ^  ' 

=  ///::(/'.  /')  +  /.■/;;(",  />)  +  ^^  +  u.-. 

Avhere  the  6"s  arc  ]>ro]»ci-  fractions,  tlie  ^"s  intiiiitcsimals. 
To  prove  the  lirst  form,  add  and  suljtract ./'('/  +  //.  h)  ;   then 

Af^lfin  +  h.  h)-f{n.  h)]  +  [f{a  +  h.  h  +  k)-fOi  +  h.  /-)] 
=  lif',\n  +  d^h.  h)  +  /,;/■;(,(  +  //.  h  -\-  OJc) 

by  the  application  of  the  Theorem  of  the  INIean  for  functions  of  a  single  \arialile 
(§§  7,  2(i).  The  api)lication  may  be  made  because  tlie  function  is  continuous  and 
the  indicated  derivatives  exist.  Now  if  the  derivatives  are  also  continuous,  tlicy 
may  be  expressed  as 


PAETIAL  DIFFERENTIATION;   EXPLICIT  95 

where  fj,  f,  may  be  made  as  small  as  desired  by  taking  h  and  k  sufficiently  small. 
Hence  the  third  form  follows  from  the  first.  The  second  form,  which  is  symmetric 
in  the  increments  h,  k,  may  be  obtained  by  writing  x  =  a  +  th  and  y  =  b  +  tk. 
Then/(j-.  ;/)  =  <{>(0-  As/  is  continuous  in  (x,  y),  the  function  4>  is  continuous  in  t 
and  its  increment  is 


A*  =  f{a  +  t  +  Ath,  b  +  t  +  Atk)  -f{a  +  th,  b  +  tk). 

This  may  be  regarded  as  the  increment  of  /  taken  from  the  point  {x,  y)  with  At  •  h 
and  At  ■  k  as  increment-;  in  x  and  y.    Hence  A*  may  be  written  as 

A*  =  Af  ■  ///;;  (a  +  ih,  b  +  tk)  +  At  ■  kf'^  {a  +  th,  b  +  tk)  +  fj A<  •  h  +  ^M  ■  k. 

Now  if  A<t>  be  divided  by  At  and  At  be  allowed  to  approach  zero,  it  is  seen  that 

At  ,  ,  r/4> 

lim  —  =  h  f'  {a  +  Uu  b  +  tk)  +  k  f,  {<(  +  th,  b  +  tk)  =  —  • 

The  Theorem  of  the  Mean  may  now  be  applied  to  <l>  to  give  4>  (1)  —  t  (0)  =  1  •  i''{ff), 
and  hence 

<!>  (1)  -  *  (0)  =  fin  +  /(.  b  +  k)  -  f{n.  6) 

=  Af=hf'_^{a  +  dh,  b  +  ek)  +  kfy{,i  +  Oh,  b  +  6k). 

47.    mXm  jKirfliil  different  id  Is  of /'may  Le  defined  as 

(f ,f  =^  f',A:r,      so  that      d.r  =  \r,  -^j— =  ^  , 

a. I'        ex 

.If       cf  ('') 

(L,f=  T",A>/,     so  tliat     (Jii  =  \i/.         -''r^  =  -r-i 
"        '  ■     '  '  '  <'!J        cy 

where  the  indices  .'•  and  //  introduced  in  </,./and  f^/,,/' indicate  tliat  .r  and 
1/  respectively  are  ah)ne  allowed  to  vary  in  forming  the  corresjjonding 
partial  differentials.    The  total  differential 

'{f  =  <^rf  +  dj  =  t^  d.r  +  ?^  du,  (6) 

Avliich  is  the  sum  of  the  partial  differentials,  may  1)6  defined  as  that 
sum ;  but  it  is  better  defined  as  that  part  of  the  increment 

\f  ==  ?^  ^■'-  +  ?^  A//  +  L\.r  +  lAu  (7) 

Avhicli  is  obtained  by  neglecting  the  terms  ^^A.r  +  tA/A  "\vhich  are  of 
higher  order  than  A./'  and  \y.  The  total  differential  may  therefore  be 
computed  bv  finding  the  partial  derivatives,  multiplying  them  respec- 
tively by  dx  and  (///,  and  adding. 

The  total  differential  of  «  =  /(.r,  ij)  may  be  formed  for  (.r^,  tj^  as 

'"cf\  ,. ,  ,  (if 


,^''-^'o)  +  (|)/-'^-^o).  (8) 


where  the  values  x  —  x^  and  //  —  y^  are  given  to  the  independent  differ- 
entials '/.'•  and  '///,  and  df  =  dz  is  Avritten  as  z  —  z  .    This,  however,  is 


96 


DIFFERENTIAL  CALCULUS 


the  equation  of  a  plane  since  x  and  ij  are  independent.  The  difference 
Af—  df  whicli  measures  the  distance  from  tlie  plane  to  the  surface 
along  a  parallel  to  the  .-i-axis  is  of  higher  order  than  Va,/''-^  +  A/y- ;  for 


\f-<lf 


VA,7y^  +  Ay- 


r\.r  +  C.A// 


<!C,|  +  ILl  =  o. 


Va;/--  +  A.;/ 

Hence  the  plane  (8)  will  be  defined  as  the  tcmjcnt  plana  at  (.r^^,  y^,  z^ 
to  the  surface  z  =fQi-,  ij).    The  normal  to  the  plane  is 


% 


-1 


(^) 


1>I> 


A./ 


which  will  be  defined  as  the  nornml  to  ilia  surface  at  (.r^,  y^,  z^^.  The 
tangent  plane  will  cut  the  ])lanes  y  =  y^  and  ./•  =  x^  in  lines  of  which 
the  slope  is  f^^  and  f',j^.  The  surface  will  cut  these  planes  in  curves 
which  are  tangent  to  the  lines. 

In  the  figure,  PQSR  is  a  portion  of  the 
surface  z  =f(x,  y)  and  PT'TT"  is  a  cor- 
responding portion  of  its  tangent  plane 
at  P(x^,  y^^,  -.'q).  Xow  the  various  values 
may  be  read  off. 

I"Q  =  A,/, 
J"r  =  ,/J] 

x's  =  A/; 

48.  If  the  variaUes  x  and  //  are  expressed  as  ./■  =  <j)(f)  and  //  =  {{/(t) 
so  that /'(./•,  y)  becomes  a  function  of  t,  the  derivative^  of /'with  respect 
to  t  is  found  from  the  ex})ression  for  the  increment  of/'. 

\f       'If 


p'T'/pp'  =/;, 
p"r"/pp"  =;;, 

P'T'  +  P"T"  =  X'T, 


ct  A.''       cf  \ii       ^  A.''        <,  A// 

ex  \t       cy  \f        ^'  M         -  \t 


_,  ,,         cf  <lx       Cfily 

hm  — ■  =  ---  =  ' r  ^  '- r 

A(  =  o  At        (It       ex  ilt       cy  at 


(10) 


The  conclusion  I'cquii'es  that  x  and  //  should  have  finite  derivatives  Avith 
respect  to  t.    The  ditt'erential  of /'as  a  function  of  t  is 


(It    ^        cf  (Ix  cf  (111    ,        cf    ^         Cf    ^ 

(If  =  ^(]t  =  4 (It  +  f~^  (It  =  7^  (Ix  +  /-  (hi 

(It  ex  (It  cy  (It  ex  cy    ' 


(11 


and  hence  it  a])])eai's  that  tJ/c  (Hifcrcnt'tal  I/as  tlw  saiin'  form  as  the  total 
differential.    This  result  will  be  generalized  later. 


PARTIAL  DIFFERENTIATIOX ;   EXPLICIT 


97 


As  a  particular  case  of  (10)  suppose  that  :i-  and  //  are  so  related  tliat 
the  point  (.r,  ;/)  moves  along  a  line  inclined  at  an  angle  r  to  the  rr -ax is. 
If  s  denote  distance  along  the  line,  then 

a- =  .r^  +  *■  cos  T,     ?/ =  ?/p -f  .s- sin  T,     r/./- =  cos  rr/.s',     di/  =  auiTds    (12) 

df       dfdxdfd!,  . 

and  -r  =  ~T~  +  ~^=^/x  cos  t  +  /„  sm  r.  (13) 

ds       ox  ds       oij  ds      ^  '  ■'  ^     ^ 

The  derivative  (13)  is  called  the  dlrerflonnJ  dcrirdtlrc  of /in  the  direc- 
tion of  the  line.  The  i)artial  derivatives  f'-,.,  /',^  are  the  particular  direc- 
tional derivatives  along  the  directions  of  the  ./'-axis  and  ?/-axis.  The 
directional  derivative  of  _/'  in  any  direction  is  the  rate  of  increase  of 
/'  along  that  direction  ;  \i  z  =  f(.r,  ;/)  be  inter- 
preted as  a  surface,  the  directional  derivative  is 
the  slope  of  the  curve  in  which  a  plane  through 
the  line  (12)  and  perpendicular  to  the  .t //-plane 
cuts  the  surface.  If  /'(.r,  y)  be  represented  liy 
its  contoiir  lines,  the  derivative  at  a  i)oint 
(.r,  y)  in  any  direction  is  the  limit  of  the  ratio 
A /'/As  —  AT/A.s  of  the  increase  of/',  from  one  contour  line  to  a  neigh- 
boring one,  to  the  distance  between  the  lines  in  that  direction.  It  is 
therefore  evident  that  the  derivative  along  any  contour  line  is  zero  and 
that  the  derivative  along  the  normal  to  the  contour  line  is  greater  than 
in  any  other  direction  because  the  element  dn  of  the  normal  is  less  than 
ds  in  any  other  direction.  In  fact,  a])art  fi'om  inlinitesimals  of  higher 
order, 


& 


A?t 

—  =  cos  xb. 


A/ 
A.s 


A  f  ilf       df 

■     cos^,     -  =  -cos^. 


A/i 


ds 


(1^) 


Hence  it  is  seen  that  flw  dcrirdtlrc  (doiuj  (inij  direction  ukii/  he  found 
hy  inultvpl ijiiKj  the  deriratirc  (ilon^  tlic  noi'inal  In/  the  cosine  of  tJic  (mf/le 
between  that  direction  and  the  normol.  The  dei'ivative  along  the  normal 
to  a  contour  line  is  called  the  norniol  dcrivdtire  of /'and  is,  of  course, 
a  function  of  (.r,  ?/). 

49.  ISText  suppose  that  n  =  f(.r,  //,  --,  •  •  •)  is  a  function  of  any  number 
of  variables.  The  reasoning  of  tht>  foregoing  paragraphs  may  be 
repeated  without  change  except  for  the  additicjnal  number  of  variables. 
The  increment  of /'will  take  any  of  the  forms 

=  hf:(a  +  eji,  />,  c, . . .)  +  ff;(o  +  /^  /.  -f  ej:,  c,.--) 

+  lf:(n  +  h,  b  +  f,  e  +  BJ,  ...)-f  ... 
=  V'.C  +  kf'y  +  V:+--  •].  +  e,,.  r,  +  Ok.  r  .u  m.... 

=  kfr  +  //;  +  (r:  +  ■  •  •  +  C/  +  V^  +  1,1  +  ■■■, 


98  DIFFERENTIAL  CALCULUS 

and  the  total  differential  will  naturally  he  defined  as 

and  finally  if  a',  y,  z,  •  •  •  be  functions  of  ;*,  it  follows  that 

(It        d.r  (It       c.j  dt        cz  (It  ^    ^  ^ 

and  the  differential  of /'as  a  function  of  ;*  is  still  (16). 

If  the  variables  ./',  i/,  -:,  ■  ■  ■  Avere  expressed  in  terms  of  several  new 
variables  /•,  s,  •  ■  • ,  the  function  /'  would  become  a  function  of  those  vari- 
ables. To  iind  the  partial  derivative  of  /  with  respect  to  one  of  those 
variables,  say  /•,  the  remaining  ones,  .s,  •••,  would  l)e  held  constant  and 
/  would  for  the  moment  l)ecome  a  function  of  /•  alone,  and  so  would  ,/•, 
y,z,---.    Hence  (17)  may  be  applied  to  ol>tain  the  partial  derivatives 

cr        ex  cr       Cii  cr       cz  cr  ' 

. ,,       ...  /  .  .  ...  (18) 

Cf  Ct  cr         Cf  Cij         Cf  CZ  ^       ' 

and  T^  =  7^ -:r-  +  7^-7^  +  -7— -:—  +  ■•■.  etc. 

C.S  Cj-  Cs  C//  €.■<  CZ  Cs 

These  are  the  formulas  for  cl/'inf/e  of  rnrlahlr  analogous  to  (4j  of  §  2. 
If  these  equations  be  multiplied  In-  A/-,  An,  •  •  •  and  added, 

cf  cf  ,  cf  Ic.i-  C.I'  \       cf /cii  \ 

^  A/-  +  T^  Ax  +  •  •  •  =  ^   —  A/-  +  —  A.s-  +  •  •  •    +  7^  M  A/-  +  •  •  •    +  .  •  -. 

cr  cs  c.r  \  cr  cs  /       cij  \cr  j 

or  (If  =  7—  '/.'•  +  --  <hi  +  :;•  -  ilz  +  •  •  • ; 

cr  Cij      ■  CZ 

for  when  /■,  .s,  •••  are  the  inde})endent  variables,  the  parentheses  al)Ove 
are  cA/',  c///,  ^At',  •  •  ■  and  the  expression  on  the  left  is  df. 

Theokkm.  The  expression  of  the  total  dittVi'ential  of  a  function  of 
.V',  //,  ,--■,  •■•  as  (If  =  f'jl.r -\-  f%li/ -\- f'jiz -\- . .  ■  is  the  sanu-  whether  ./■.  //, 
'.',  •••  ai'C  the  independent  variables  or  functions  of  other  indcjjcndent 
variables  /•,  .v,  •  •  • ;  it  being  assumed  that  all  the  derivatives  which  occur, 
whether  of  /'  by  ./•,  y,  z,  ■  ■  ■  or  of  .r,  y,  z,  ■  ■  ■  by  r,  s,  ■  ■  ■,  are  continuous 
functions. 

By  the  same  reasoning  or  ly  virtue  of  this  theorem  the  rules 

(I  (riA  =  r/I/i^      il  { II  -\-  r  —  //•)  =  i1  II  -\-  il r  —  (■///•, 

7/                    ,      ,        ,            J"\        rihi  -  iiilr  (19) 

,1  {-,,,■)—  mil'  ^  nhi.      d[-\=- ,  ^      ^ 


of  the  diffei'cntial  calculus  will  apply  to  calculate  the  total  differential 
of  combinations  oi'  functions  of  several  variables.  If  In'  this  means,  or 
any  other,  tliere  is  olitaiiied  an  expression 


PARTIAL  DIFFEREXTIATIOX  ;   EXPLICIT  99 

df  =R(r,s,f,...)  dr  +  S{,;s,t,...)  d,  +  T  (r,  s,  f,  .  .  .)dt -{- ■  ■  ■    (20) 

for  the  total  differential  in  Avliieh  r,  s,  t,  ■  ■  ■  are  Inde^iendt'nt  variables, 
the  coefficients  R,  S,  T,  ■  ■  ■  are  the  derivatives 

R  =  ^,     S  =  '-l,     T  =  ¥-,....  (21) 

Cj-  Cs  Cf  ^       ^ 

For  in  the  equation  df^  Rdr+Sds-j-  Tdt -\ =f;.dr-{-flds^f;dt-\ , 

the  variables  r,  s,  f,  ■  ■  ■,  Ijeing  independent,  may  be  assigned  increments 
absolutely  at  pleasure  and  if  the  particular  choice  dr  =  1,  d.'i  =  df  =  ---  =  0, 
l)e  made,  it  follows  that  /.'  =./',':  and  so  on.  Tlie  single  equation  (20)  is 
thus  equivalent  to  the  e(|uations  (21)  in  number  e(|ual  to  the  number  of 
the  independent  variables. 

As  an  example,  coiisiiler  the  case  of  the  function  tan-i  (y/x).    By  the  rules  (1!»), 
,;  f^j^- 1  y  _     '-l(y/-f)     _  '^y/'>^  —  yds/x-  _  j-dy  -  ydx 

X  ~  1  +  ((///)-  ~     1  +  ((///)-     "    Jt-  +  r 

Then  ^tan-i?^= ^^ — ,         -^tan-i^  =  -^ ,         by  (20)-(21). 

ex  X  X-  +  y-  cy  x      x-  +  y'- 

If  y  and  x  were  expressed  as  y  =  siidi  rst  and  x  =  cosh  rst,  then 

_  1  .y      ■'''^y  ~  y'-^"''       {^t-dr  +  rtdn  +  r^idf]  [cosh-r.s<  —  sinh-rs^] 

cosh'-'/'.si  +  sinh-/>-i 
cf  rt  cf  rs 


X         X-  +  y- 

and 

cf  _         .s( 
cr       cosh  2  r.st 

•s       cosh  2  rst  ct       cosh  2  rst 

EXERCISES 

1.  Find  the  partial  derivatives/,',  f'^  'ir/_^'.  f^.  /.'  of  these  functions  : 
{a)  loiiix- +  y").  (/3)  e-^  ens  y  sins:.  (7)  x- +  3  xy  + //', 

(5)  -^,  (e)  ^--^^,  ii-)  ln,.(sinx  +  sin^y  +  sin^^). 

(.)  sin-i^  (0)^ei;  (0  tanh-^V2("^^  +  4  +  ^^y. 

^"  X  ^  '  X  \x-  +  y-  +  z-/ 

2.  Apply  tlie  definition  (2)  directly  to  the  following  to  find  the  partial  deriva- 
tives at  the  indicated  points  : 

(a)  ~j  at  (1,  1),  (p)  X-  +  3xy  +  r  :it  (0,  0).  and  (7)  at  (1,  1), 

(5)  5^^^  at  (0.  0):  also  trv  differentiating;-  and  suVistitutiim-  (0,  0). 
^    '  X  +  y       ^        '  -  -  ■     ■       ' 

3.  Find  the  partial  derivatives  and  hence  the  total  differential  nf  : 


e-n/ 


(a)  -^^ — ~^,  (/3)  xlog.yz,  (7)  V<(- 


X-  +  y- 


y-- 


(0)  f-^siny.  (e)  e=^-sinhx?/,  (j-)  log  tan/x  + -j  ;/j, 


yV  ,r.\  ^  ~"  ?/  l-'ix  _    _\,    .  z-x- 


W       .  W^.  (.)i"=Uv  +  \'  +  ^ 


x^z 


100  DIB^FEREXTIAL  CALCULUS 

4.  Find  the  general  eciuations  of  the  tangent  plane  and  normal  line  to  these 
surfaces  and  find  the  eijuations  of  tlie  plane  and  line  for  the  indicated  (x^,  y^^)  : 

(a)  the  helicoid  z  =  /c  tan- i  (y/x),  (1,  0),  (1,-1),  (0,  1), 

(/3)  the  paraboloid  4pz  =  jx"  +  y-),  (0,  p),  (2p,  0),  {p,  —  p), 

(y)  the  hemisphere  z  =  Va-  —  x-  —  y-,  (0,  —  J  «),  {\  a,  I  a),  (l  Vs  «,  0), 

(5)  the  cubic  xyz  =  1,  (1,  1,  1),  {- i,  -  h  ^i  (4,  h  D- 

5.  Find  the  derivative  with  respect  to  t  in  these  cases  by  (10) : 

((If)  f  —  x"  +  y",  X  =  a  cos  t^  y  =  h  sin  i,     (/3)  tan-  ^  \/  ;  ^  y  —  cosh  t,  x  =  sinh  i, 
(7)  sin- 1  (x  —  ?/),  X  =:  3  i,  ?/  =  4  f',  (5)  cos  2  xv/,  x  =  tan-  ^  t,  y  =^  cot-  1  i. 

6.  Find  the  directional  derivative  in  tlie  direction  indicated  and  obtain  its 
numerical  value  at  the  points  indicated  : 

{a)  x'-//,  T  ^  45^,  (1,  2),  {13)  i^m"-xy,  r  =  00^,  (V-S,  -  2). 

7.  (a)  Deternune  the  niaxinuun  value  of  df/ds  from  (13)  by  regarding  t  as 
variable  and  applying  the  ordinary  rules.  Show  that  tlie  direction  that  gives  the 
maxinuun  is  ,  r 

T  =  tan-  1    -.        and  then         —  =  \     —  )  +  U,—  I  • 
/;  dn       \\cxj       \cy] 

(/3)  Siiow  that  tlie  sum  of  the  S(iuares  of  the  derivatives  along  any  two  perpen- 
dicular directions  is  the  same  and  is  the  s(puire  of  the  normal  derivative. 


8.  Show  that  (/;  +  ?/7;')/v  1  +  y'-  and  (/;?/'  -/;)/Vl  +  y'-  are  the  deriva- 
tives of /along  the  curve  y  =  ^(x)  and  normal  to  the  curve. 

9.  If  df/dn  is  delined  by  the  work  of  Ex.  7  (a),  pn)ve  (14)  as  a  consecpience. 

10.  Apply  the  formulas  for  the  change  of  variable  to  the  following  cases  : 

(a)  r  =  Vx^+V^  0  =  tan-i^.  Find  '■^-,   ^,   x[('-'-)\  PY' 

•<;  ex     cy      \\Zxl        Vy/ 

(/3)  x  =  rcos0,  ?/  =  rsiii0.  Find    — ,  -'-,   (  '-]  -{ — -{^]  • 

cr     c(j>     Xcrf        r~\d(p/ 

(7)  X  =  2  r  —  ; J  .s-  +  7,  ?/  =  —  r  4-  8  .s  —  0.     Find      -  —  4x  +  2y  if  u  —  x-  —  y'^. 

,^,    fx  =  x' coscr  —  ?/'sin  (ir,  „,  /f/\-     /<J'\'      /^/V"  ,   /('fV 

5    -;  ,    .  ■',  Show    —     +:=-;    +    .--    • 

(,!/  =  ■''   sina •+  //  cost!-.  \f)x/       \( y /       \(x/        \cy/ 

(e)  Prove  '-L  +  'l  =  0     if    /(«,  1-)  r=f{x  -  v/,  //  -  x). 

(X       cy 

(f)  Let  X  =  dx'  +  ^.v'  -1-  i-z',  y  =  a'x'  +  h'y'  +  r'z'.  z  —  n"x'  +  //'//'  +  ''"2',  wliere 
a,  />,  c,  <('.  I/,  r',  rf",  //',  <•/'  are  tin;  direction  cosines  of  new  rectangular  axes  with 
resx)e<'l  to  the  old.  This  transformation  is  called  an  (irUux/diial  lr<insfotinati.()n.   Sliow 

11.  Define  directional  derivative  in  space  ;   also  normal  derivative  and  estab- 
lish (14)  for  this  case.    Find  the  normal  dei'ivative  of  /  =  xyz  at  (1,  2,  i>). 

12.  Find  the  total  diftVrenliid  and  hence  the  partial  derivatives  in  Kxs.  1,  8,  and 
{a)  h>y;{x- +  y- +  Z-),  (i^)  i/Z-c,  {-y)  x-yfi'\  {5)  xyz  loi^ xyz, 


PARTIAL  DIFFERENTIATIOX ;   EXPLICIT  101 

(e)  u  —  X-  —  y-,  X  =  rcotist,  y  =  .ssinrL  Fiiul  cu/cr,  c«/(.s,  cu/ct. 

( f )  u  =  y/x,  X  =  r  cos  (p  sin  9,  y  =  r  sin  <p  sin  $.      Find  u/,  » /,  iig. 
(77)  u  =  e^'J,  X  =  log  vr-  +  .s-,  y  =  tan-  1  (s/r).       Find  w/,  w^'. 

lo.  It  —  =    -  and  —  = ,  show  —  =  — ^  and = if  r,  <p  are  polar 

ex      ly  cy  ex  cv      r  c<p  r  c<p  Zr 

coordinates  and/,  g  are  any  two  functions. 

14.  If  p{x,  y,  z,  t)  is  the  pressure  in  a  fluid,  or  p(x,  ?/,  z,  t)  is  the  density,  depend- 
ing on  the  position  in  the  fluid  and  on  the  time,  and  if  7<,  ?),  w  are  the  velocities  of 
the  particles  of  the  fluid  along  the  axes, 

dp         cp         cp  cp      cp  ,     dp  cp         (o  CO      (p 

dt         ix         cy  cz       It  dt         ex         cy  cz       dl 

Explain  the  meaning  of  each  derivative  and  prove  the  formula. 

15.  If  z  =  xy,  interpret  z  as  tlie  area  of  a  rectangle  and  mark  dxZ,  AyZ.,  Az  on  the 
figure.    Consider  likewise  u  =  xyz  as  the  volume  of  a  rectangidar  parallelei)iped. 

16.  Small  errors.  If  /(x,  y)  be  a  (juaiitity  determined  by  measurements  on  x 
and  2/,  the  error  in  /  due  to  small  errors  dx,  dy  in  x  and  y  may  be  estimated  as 
df  =  f^dx  -\-  fydy  and  the  relative  error  may  be  taken  as  df  -^f  —  dlogf.  Why 
is  this  ? 

(a)  Suppose  S  ■=  I  ah  am  C  be  the  area  of  a  triangle  with  a  =  10,  h  =  20,  C  =  30"^. 
Find  the  error  and  the  relative  error  if  a  is  subject  to  an  error  of  0.1.   Ans.  0.5,  1%. 

(P)  In  (tr)  suppose  C  were  liable  to  an  error  of  10'  of  arc.  Ans.  0.27,  |%. 

(7)  If  (I,  6,  C  are  liable  to  errors  of  f/^',  the  cond)ined  error  in  S  may  be  3.1%. 

(5)  The  radius  r  of  a  capillary  tube  is  determined  fn)m  13.()7rr-/  =  iv  by  find- 
ing the  weight  iv  of  a  colunni  of  mercury  of  lengtli  I.  If  w  =  1  gram  with  an  err(u- 
of  10-^  gr.  and  I  =  10  cm.  with  an  error  of  0.2  cm.,  determine  the  possible  error 
and  relative  error  in  r.  Ans.  1.05%,  5  x  10-  ',  mostly  due  to  error  in  I. 

(e)  The  fornmla  c'  =  0-  -|-  ?/-  —  2r//jci)s  (.'  is  used  to  determine  c  where  a  =  20, 
b  =  20,  C  =  00°  with  possible  errors  of  0.1  in  a  and  b  and  30'  in  C.  Find  the  possible 
absolute  and  relative  errors  in  c.  Ans.  1,  1|%. 

(f)  The  possible  percentage  error  of  a  product  is  the  sum  of  the  percentage 
errors  of  the  factors. 

(7/)  The  constant  <j  of  gi'avity  is  determined  from  y  =  2 si--  by  observing  a  body 
fall.  If  .s'  is  set  at  4  ft.  and  (  detenuined  at  about  '  sec,  show  that  the  error  in  r/ 
is  almost  wholly  due  to  the  error  in  /,  that  is,  that  s  can  be  set  very  nuich  more 
accurately  than  t  can  be  determined.  For  example,  find  the  error  in  I  which  would 
make  the  same  error  in  g  as  an  error  of  J  inch  in  .s. 

(ff)  The  constant  g  is  determined  by  gt-  =  ir'-l  with  a  pendulum  of  lengtli  I  and 
period  t.  Suppose  t  is  determined  by  taking  the  time  100  sec.  of  100  beats  of  the 
pendulum  with  a  stop  watch  that  measures  to  1  sec.  and  that  I  may  be  measured 
as  100  cm.  accurate  to  l  millimetei'.    Discuss  the  errors  in  g. 

17.  Let  the  coordinate  x  of  a  particle  be  x  =f{q^,  (/.,)  and  depend  on  two  inde- 
pendent variables  q^,  q.-,.    Show  that  the  velocity  and  kinetic  energy  are 

''=.C'^f +4'^''  T=lmf-  =  a^^,yl+  2a,Ji^i^  +  a^Jil 


102  DIFFEREXTIAL  CALCULUS 

where  dots  denote  differentiation  by  t,  and  a^j,  a^g,  ^02  ^^'^  functions  of  {q^,  g.,). 

Show  —  =  — ,  z  =  1.  2,  and  similarly  for  any  number  of  variables  q. 
cqi      cQi 

18.  The  helix  x  =  a  cosi,  y  =  a  sin  t,z  =  at  tan  a  cuts  the  spliere  x'^  +  y'^  +  z^  — 
«^sec-^  at  sin-i  (sin  cr  sin/3). 

19.  Apply  the  Theorem  of  the  Mean  to  prove  that  /(x,  y.  z)  is  a  constant  if 
f^  =fy  =f^  =  0  is  true  for  all  values  of  x,  y,  z.  Compare  Theorem  10  (§  27)  and 
nialce  the  statement  accurate. 


20.  Transform  ^  =  A/'i^)  +  (^)  +  (^)  to  (a)  cylindrical  and  (/3)  polar 
coordinates  (§  40). 

21.  Find  tlie  aui^le  of  intersection  of  the  helix  x  =  2cosi,  y  =  2  sin  ^  z  =  t  and 
the  surface  xyz  =  1  at  their  lirst  intersection,  that  is,  with  0  <  i  <  1  tt. 

22.  Let/,  y,  h  be  three  functions  of  (x,  y,  z).  In  cylindrical  coordinates  (§  40) 
form  tlie  combinations  F  =  /cos  (p  +  g sin  cp^  G  =  —  /sin  <p  +  y  cos  0,  II  =  h.  Trans- 
form .J..         _,  .,,_  -         ^j. 

(a)^  +  ^  +  ~,  {/3)^-^>  (7)^-^ 

cx       cy       c2  cy       cz  cx       cy 

to  cylindrical  coordinates  and  express  in  terms  of  F.  G,  II  in  simplest  form. 

23.  Given  the  functions  y^  and  (z'')^  and  z<-^^).  Find  the  total  differentials  and 
hence  obtain  the  derivatives  of  x-'-  and  (x-*)^  and  x(' '>. 

50.  Derivatives  of  higher  order.  If  the  first  derivatives  be  again 
differentiated,  there  arise  four  derivatives  /^'.,  /','^,  f'J,.,  /',^^  of  the  second 
order,  where  the  first  subscript  denotes  the  tirst  differentiation.  Tliese 
may  also  be  written 

where  the  derivative  of  cfjcij  with  respect  to  ,/■  is  written  c-f.'c.r'ci/ 
with  the  variabk'S  in  the  same  order  as  required  in  yv,,/v,^ /'and  opposite 
to  the  order  of  the  subscripts  in/^j..  This  matter  of  order  is  usutilly  of 
no  importance  owing  to  the  theorem:  If  tint  (hn-lcdtlri's  f'^..  f^^  Junw 
diu'lc<dli't's  f',',^,  f',^\.  (rliidi  are  contbuiotis  in  (,/'.  //)  in  tJw  ncijJilini-JiiKKl 
of  (i.nij  jinint   (.'\^.    i/^f   tlie   derlratu-cs  f"^,j   anil  fy^.   are   i-'jiial,   tliat   is. 

The  theorem  may  l>e  proved  by  repeated  application  of  the  Theorem  of  the 
Mean.    For 

[/(Xq  +  h.  y„  +  k)-f{x,,.  y,  +  k)]  -  [/(x„  +  h.  2/,)-/(x,,  //,,)]  =  V'PilU.  +  /'•)-  0(.V„)] 
=  [/K+^'-  //o  +  /^-)-/K+^^  .'/„)]- [/(Ar  //o  +  ^O-ZK-  .'/o)j  =  [-f(''-o  +  ^')--y^U-o)J 
where  0(//)  stands  for  /(x^  +  h.  y)~f{x^^.  y)  and  4,{x)  for  /(x,  //„  +  A')  -/(^,  V^)- 
Now 

<p{y,^  +  k)  -  ,p(i/,)  =  k4>'(y,,  +  dk)  =  A-[/;(x„  +  h.  //,,  +  dk)-f;,(-r,r  y,,  +  ^'n]. 

^ (.Co  +  '0  -  -^  (.'-u)  =  /'■Vi-';  +  0'/')  =  f' [/; (^0  +  ff'f'-  y.  +  ^■)  -/. (''-o  +  ^'^'-  y.)] 


c-f 

cY 

t..r    = 

—   „            J 

^,„ 

^■'■^// 

^//" 

PARTIAL  DIFFEKE^TIATIOX;   EXPLICIT  103 

by  ai:)plyiiig  the  Theoreni  of  the  Mean  to  (p{i/)  and  \p{x)  regarded  as  functions  of  a 
single  variable  and  then  substituting.  The  results  obtained  are  necessarily  equal 
to  each  other  ;  but  each  of  these  is  in  form  for  another  application  of  the  theorem. 

Hence  fy',:(:'^o  +  '?/':  ^o  +  Olc)  =.C,{''-o  +  ^''^N  Vo  +  '/'^')- 

As  the  derivatives  /J'.,  /^'^  are  supposed  to  exist  and  be  continuous  in  the  variables 
(x.  y)  at  and  in  tlie  neighborhood  of  (x^^,  y^^).  the  limit  of  each  side  of  the  ecjuation 
exists  as  /l  =  0,  k  =:  0  and  the  equation  is  true  in  the  limit.    Hence 

The  differentiation  of  the  three  derivatives/^^, /''.'^  =y'J^., _fj'^  Avill  give 
six  derivatives  of  the'  third  order.  Consider  /',','^,  and  /"^'.  Tliese  niay 
be  written  as  {f^ )',',,  and  (f',)y[r  aiul  are  equal  by  the  theoreni  just  proved 
(provided  the  restrictions  as  to  continuity  and  existence  are  satisfied). 
A  similar  conclusion  holds  for  /y,'^,  and  f',',^.^ ;  the  nuruljer  of  distinct 
derivatives  of  the  third  order  reduces  from  six  to  four,  just  as  the 
number  of  the  second  order  reduces  from  four  to  three.  In  like  manner 
for  derivatives  of  any  order,  flit'  rahie  of  flic  (Ivrirdfln^  ili'pi'uds  not  on 
tlie  order  In  vlih-li  thr  indirldaal  dljft'rcntlotlon.'i  irlfli  respi'cf  to  ./■  and 
II  are  jierfornied,  hat  onlij  on  the  total  narnher  of  d'ljferenthftlons  trlth 
resjierf  to  eocli,  and  the  result  niay  be  written  with  the  differentiations 
collected  as  -,„-^„  . 

J).^J-)n  f  ^  ^1^  ^    fOn  -  .)      ^^^._  /22) 

Analogous  results  liold  for  functions  of  any  number  of  variables.  If 
several  derivatives  are  to  Ije  found  and  added  togetlier,  a  symbolic 
form  of  Avriting  is  frequently  advantageous.    For  example, 

.     .         ,  .  c''f  cf'f 

or         (T>,  +  T)^ff  =  (j>^  +  2  Dj)y  +  iy^)f  =  fz  +  2/:;  +  /;;. 

51.  It  is  sometimes  necessary  to  rlt(/n;je  t/ie  roriotde  in  higher  deriv- 
atives, particularly  in  those  of  the  second  order.  This  is  done  by  a 
repeated  application  of  (18j.  Thus  /',".  "^vould  he  found  Ijy  differentiat- 
ing the  first  equation  with  respect  to  /•,  and  f',.^  1)V  differentiating  the 
first  l)y  .s-  or  the  second  by  r,  and  so  on.  Compare  p.  12.  Tlie  exercise 
below  illustrates  the  metliod.  It  may  be  remarked  that  the  use  of  hif/I/ej- 
dijferentio/s  is  often  of  advantage,  although  these  differentials,  like  the 
higher  differentials  of  functions  of  a  single  variable  (Exs.  10,  16-19, 
p.  G7),  have  the  disadvantage  that  their  form  depends  on  what  the 
independent  variables  are.  This  is  also  illustrated  below.  It  should  l)e 
particularly  borne  in  mind  that  the  great  value  of  the  first  differential 


104  DIFFERENTIAL  CALCULUS 

lies  in  the  facts  that  it  may  be  treated  like  a  finite  quantity  and  that 
its  form  is  independent  of  the  variables. 

To  change  the  variable  in  v^  +  i^'^  to  polar  coordinates  and  sliow 

c'-v       c'-v  _  c-v       lev       1  c'-v  (x  =  rcoii(p,  y  =  rm\(p. 

ex-       cy-       cf-       rcr       r- ccp'-'  I  r  =  Vx-  +  y-,     (p  =  Uu\- '^  {y / x) . 

cv       ev  cr       cv  cd>  cv       ev  cr       cv  c4> 

Then  —  = h—  — .         —  = V 

ex      cr  ex      c<p  cx  cy      cr  cy      c<p  cy 

hy  applying  (18)  directly  with  x.  y  taking  the  place  of  r,  .s,  •  •  ■  and  r,  (f>  the  place 
of  X.  y,  z,  •  ■  ■ .    These  expressions  may  be  reduced  so  that 


cv 

cv         X              cv      —  ;/ 

cv  X        cv  —  } 

cx 

cr  Vj-  +  y-       c4>  X-  +  y- 

cr  r       c(p    r- 

c-v 

c  cv        c  cv     cr        c    cv 

c4> 

—  z^   — .  -U  

ex- 

cx  cx       cr  cx    cx       c0  cx 

cx 

Next 

cx- 

[c-v X       cv  c  X        c'-v   —  y       cv   c   —  ?/nx 
cr- r       cr  cr  r      creep    r-         ccpcr    r-  J  r 

[c'-v    X        cv    C   X         c'-v  —  II        ff    f    —  ?/!—  w 
-^-  -  +  ^  --  -  +  -  ,  ^^  +  —  —  -    ,    ^  • 
ccpcrr      creep  r       c(p-    r-        ccpccp    r-  J    /- 

The  differentiations  of  x/r  and  —  y/r'-  may  bf  performed  as  indicated  with  respect  to 
/■,  0,  remembering  tliat,  as  r,  <p  are  independent,  the  derivative  of  r  h\  cp  is  U.    Then 

c'-v  _x'- c'-v      y'- cv      _^xy  c'-v    .   ./^U  (^^'       !/- e'-v 
(./■■-       ;■'-  c r'-       H  cr  r^  creep  r'  c(p       r^  erp'- 

In  like  manner  c'-v/ey'-  may  be  found,  and  the  sum  of  the  twn  derivatives  reduces 
to  the  desired  expression.    This  method  is  long  and  tedious  thouLili  straigiitforward. 
It  is  considerably  shorter  to  start  with  the  expression  in  pohir  coordinates  and 
transf(jrm  l)v  the  same  method  to  the  one  in  rectan^rular  coordinates.    Thus 


ev  cX       cvci/       eV  cv    .  l/ty  ev 

1 '-      =      -    -    cos    (p     -\ sill    (p     =     ~i         -      £     -\-  ; 

cx  c  r       eycr       cx  cy  r\cx         ey' 


c'-v  c'-v     .       \         /  e'-i  c'-v    .        \         cv  cv    . 

— -^cos0+     sin0   .r  +  i cos  0  +  - ---siii^  w/ +     -cos0H sin0, 

cx'-  cycx  j        ^Ix.ey  cy'-         ]        ex  ly 

cV  cx       ev  ell  cv       .  ev  cv  cv 

1- = r  sin  0  +  -  -  /•  cos  <p  —  —       y  -\ X', 

f .r  (0       iy(<P  <-J'  ([I  (x'        cy 


1  e'-v        I'e-v    .  c'-v  \       ,    /        e-v 

-    ^=         ,>in0 COS0    //+     

r  c(p-       \cx-  f //fx  /  \      cxe y 


COS0 sill  0. 

(.',■  cy 

e  I    cv\       1  e'-v       /e'-v       e'-v\ 
1  lien  _,■-_+ ^^lj_  .._    ,. 

cr\    erf        rc<p-       ^cx-       cy-/ 

e'-v       e'-v       lei   cv\        1   c'->-       c'-v       1  ev        1   e'-v  ,^^. 

or  — -  +-=--/•-       +  —  -^,  =  -  --,  -{ \-  —.--,•  (-•') 

cX-       cy-       rer\    er;        r- eep'-       er-       rer       r'-c(p'- 

The  definit'>iis  <l;f=fZ'lx-.  d_,ayf  =  f'/^^iUdy.  il'lf  =  f !,'„<h.r-  would  naturally  be 
given  iov  jiiirtuil  dlffcnntuii-^  of  tin-  sicunil  onhr.  each  of  which  would  vanish  if,/' 
reduced  to  either  of  the  independent  ^;^■iables  ./■.  y  or  to  any  linear  fuiietion  of 
them.    Thus  the  second  differentials  of  the  indei)endeiit  variables  are  zero,    'f'he 


PARTIAL  DIFFERENTIATION;   EXPLICIT  105 

second  total  differential  would  be  obtained  by  differentiating  the  first  total  differ- 
ential. 

d^f  =  ddf  =d(^-^  dx  +  ^  dy)  =  d  —  (/x  +  d  —  dy  +  "^  d^x  +   -  d'^y  ; 
\cx  cy      j         ex  cy  ex  cy 

,    ^  ,cf       c-f  ,      ,     c-f    ,  ,c/         c-f  c-f 

but  d  -^  =  ^-  dx  H dy,         d  —  = dx  -| dy, 

cx      cx-  cycx  cy      cxZy  ty- 

and  dy  =  —^  dx^  +  2  —  dxdy  +  —^  dy'-  +  ^  a\r  +  ~  dhj.  (24) 

cxr  dxdy  dy-  cx  cy 

Tlie  last  two  terms  vanish  and  the  total  differential  reduces  to  the  first  three  terms 
if  X  and  y  are  the  independent  variables  ;  and  in  this  case  the  second  derivatives, 
f^ifxyifyy^  are  the  coefficients  of^dx-,  2  dxdy,  dy'^,  which  enables  those  derivatives 
to  be  found  by  an  extension  of  the  method  of  finding  the  first  derivatives  (§  49). 
The  method  is  particularly  useful  when  all  the  second  derivatives  are  needed. 
The  problem  of  the  change  of  variable  may  now  be  treated.    Let 

70  c'-«     ,0,0  ^'^     7       ,  ,      f '■"     7     o 

d-y  =  —  dx-  +  2  —  dxdy  -\ ^  dy- 

cx-  cx-  cy- 

C-V   ,    ,         ^    C-V     ,    ,  c'-V    ,     ,         (V   ,,  cv    ,, 

=  —  dr-  +  2 drd(p  -\ ^  d(p-  +       d-r  -\ d-(p, 

cf-  creep  c(p-  Zr  c(p 

where  x,  y  are  the  independent  variables  and  )-,  (p  other  variables  dependent  on 
them  —  in  this  case,  defined  by  the  relations  for  polar  coordinates.    Then 

dx  =  cos  (pdr  —  r  sin  <pd(p,        dy  =  sin  cpdr  +  r  cos  0d0 
or  dr  =  cos  0dx  +  sin  4)dy,         rdcp  =  —  sin  (pdx  +  cos  <pdy.  (2.5) 

Then  d-r  =  (—  sin0dx  +  cos  (pdy)d<p  =  rd<pd(p  =  rdcp-, 

drd<p  +  rd-cp  =  —  (cos  (pdx  +  sin  <pdy)  dcp  =  —  drd(p, 

where  the  differentials  of  dr  and  rd(p  have  been  found  subject  to  d-x  =  d'-y  =  0. 
Hence  d^r  =  rdep-  and  rd-(p  =  —  2drd<p.  These  may  be  substituted  in  d-y  which 
becomes 

d-v  =  —^  dr-  +  2 )drd<p  +  {  — ;  +  ''    -)d(p-. 

cf-  \crccp       r  l<pj  \Z(p'-         Zr/ 

Next  the  values  of  dr-,  drdcp,  d<p-  may  be  substituted  from  (2-3)  and 

,.,        VZ-v       ,-,         2  /  Z-v        1  Zv\  .         ,    /Z-v    ,      fLAsin-c^n  ,  „ 

d-v  =    —  cos-<p cos  (p  sin  <p  +    — ;  +  r  —  1  — ^~    dx- 

tZr-  r\crZ(p       r  Zcpl  \Z(p-         Zr/     r-    J 

rZ'-v  .  /  Z-n        1  f  cX  cos-</)  —  sin-<i       f-f  ens0sin  01 

+  2     —  cos  0  sin  0  + ^ ^  -  --  -    --r-  -^ 

\_er-  vrcKp       r  c<p/  r  c<p-         r-        J 


dxdy 


[Z-v   .    ,         2  /  Z-v        1  Zi-\  .  (Z-v         fr\cos-(;&] 

— ^  sni-0  + cos  0  sni  0  +    -,—  +  ''  —   — ~ 
cr-                r\erc(p       r  ccpj  \c<p-         cr/     r-    J 


dy-. 


Thus  finally  the  derivatives  v'^'^.  r,'^.   i-J,'^  are  the   three   brackets   wliich   are   the 
coefficients  of  dx-,  2  dxdy,  dy-.    The  value  of  v^.'^.  +  rj,^  is  as  found  before. 

52.   The  condition  /,'^=/',^,'  Avhieli  sul)sists  in  accordance  with  tlie 
i'undaniental  theorem  of  §  50  gives  tJir  ctiniJithin   ilmt 

M{.r,  !/)dx  +  X{.r,  ;/)(/;/  =  5^  <I.r  +  ^f^  ,1 H  =  df 

CX  cy 


lOG  DIFFERENTIAL  CALCULUS 

he  the  total  differential  of  some  fnnrtlon  /{x,  //).    Jnfact 
c  cf      cM       cX  _   c  cf 

Cij  C.r  Cij  C.I'  C.I'  Cij 

cM       cX  (<IM\        /'IX\ 

and  —  =  ^     or        —-     =     —-•  (2G) 

cu         c.r  V'^yA      \<l'>- Ju 

The  second  form,  where  the  variables  which  are  constant  during  tlie 
differentiation  are  explicitly  indicated  as  subscri})ts,  is  more  common  in 
Avorks  on  thermodynamics.  It  will  be  proved  later  that  conversely  if 
this  relation  (2C)j  holds,  the  expression  Md.r  +  Xdi^/  is  the  total  ditt'er- 
ential  of  sonie  function,  and  the  nietliod  of  finding  the  function  will 
also  be  given  (^^j  92,  124).  In  case  Md.r  -\-  Xdij  is  the  differential  of 
some  function /'(./•,  //)  it  is  usually  called  an  cxdrt  dfferrnfia/. 

The  ap})lication  of  the  condition  for  an  exact  differential  may  1)e 
made  in  conneetiou  with  a  problem  in  thermodynamics.  Let  .S'  and  U 
Ije  tlie  entropy  and  energy  of  a  gas  or  vapor  inclosed  in  a  receptacle  of 
volume  r  and  subjected  to  the  pressure  ji  at  the  temperature  7'.  The 
fundamental  equation  of  thermodynamics,  connecting  the  differentials 
of  energy,  entrojy,  and  volume,  is 

.!U=r,,S^j,.lr;      and      ('f)  =  -(;;^,)_  (27) 

is  the  condition  that  dU  be  a  total  differential.  Now,  any  two  of  the 
five  quantities  U,  S,  r.  T.  p  may  l)e  taken  as  indtqiendent  v;iri;iblcs.  In 
(27)  the  choice  is  -S',  r  \  if  the  equation  were  solved  for  dS.  the  choice 
woiild  be  U,  r\  and  /',  .s'  if  solved  for  ///•.  In  each  case  thi'  cross  differ- 
entiation to  exjirt'ss  the  condition  (20)  would  give  rise  to  a  relation 
between  the  derivatives. 

If  J).  T  were  desired  as  independent  variables,  the  clianue  of  variable 
should  be  made.    The  expressiun  of  the  ennditinii  is  then 

'  ^0,-"Oi\\r  {.)!.[''(^r)rK-rrl]}. 

/'^'^■\         ,,.    (^  (-V    _        (-S         /(Zi'\  <-c 

^dp  I  r'^      irip  "  ''  f-  T(j>  ~      cpcT  ~  VlT  \r  ^'  (pcT  ' 


^  1' 


where  the  differentiation  (ju  the  left  is  made  with  p  mnstant  and  that  nn  tlie  rii:ht 
with  7'  cnnstant  and  where  tlie  .-ubsci'iiits  liave  been  dniijped  frnm  the  sreimil 
derivatives  and  the  usual  notatinn  adopted.  l''.\erythiny  cancels  exce^jt  two  terms 
whit'h  uive 


PAKTIAL  DIFFEREXTIATIOX;   EXPLICIT  107 

C^^-m       or     i/^)=-(i''^\.  (28) 

Xdpjr         \dTJr.  T\dp/r         \dT/p  ^     > 

The  importance  of  tlic  tt'st  for  an  exact  differential  lies  not  only  in  the  relations 
obtained  between  the  derivatives  as  above,  but  also  in  the  fact  that  in  applied 
mathematics  a  great  many  expressions  are  written  as  differentials  which  are  not 
the  total  differentials  of  any  functions  and  which  nnist  be  distinnuished  from  exact 
differentials.  For  instance  if  dll  denote  the  infinitesimal  portion  of  heat  added 
to  the  i,^as  or  vapor  above  considered,  the  fundamental  equation  is  expressed  as 
dll  —  dU  +  jidv.  That  is  to  say,  the  amount  of  heat  added  is  e(jual  to  the  increase 
in  the  eneri^y  plus  the  work  done  by  the  sas  in  expanding.  Now  dll  is  not  the  dif- 
ferential of  any  function  H{C,  v)  ;  it  is  dS  =  dll/T  which  is  the  differential,  and 
this  is  one  reason  for  introducing  the  entropy  .S.  Again  if  the  forces  A",  Y  act  on  a 
particle,  the  irork  done  during  the  displacement  through  the  arc  f/.s  =  Vdx-  +  di/'- 
is  written  dW  =  Xdx  +  Yd)/.  It  may  happen  that  this  is  the  total  differential  of 
some  function  ;   indeed,  if 

d]V=-dV{x,y).     A'd.r  +  Ydy  =  -d V,     A'  =  -  —  ,     1'  =  -  -- , 

ex  cy 

where  the  negative  sign  is  introduced  in  accordance  with  custom,  the  function  V  is 
called  the  jjoientiiil  energy  of  the  particle.  In  general,  however,  there  is  no  poten- 
tial energy  function  T.  and  dW  is  not  an  exact  differential  ;  this  is  always  true 
when  part  of  the  work  is  due  to  forces  of  friction.  A  notation  which  should  dis- 
tinguish between  exact  differentials  and  those  which  are  not  exact  is  much  more 
needed  than  a  notation  to  distinguish  between  partial  and  ordinary  derivatives; 
but  there  appears  to  he  none. 

Many  of  the  ])liysical  magnitudes  of  thermodynamics  are  expressed  as  deriva- 
tives and  such  relations  as  (2(i)  estaljlish  relations  between  the  magnitudes.  Some 
definitions  : 

specific  lieat  at  constant  volunu;  i> 

specific  lieat  at  constant  pressure  i:- 

latent  heat  of  expansion  i.- 

coefficient  of  cubic  expansion  i.^ 

modulus  of  elasticit}'  (isothermal)  i.- 

modulus  of  elasticity  (adiabatic)  i.- 

53.  A  [lolyiiuniitil  is  stiid  to  1)0  liomogcneous  wlieii  eacli  of  its  terms 
is  (if  tile  same  order  wlicii  all  the  vari;il)les  are  considered.  A  defini- 
tion of  homogeneity  which  includes  this  etise  and  is  ap})lieable  to  more 
general  etises  is  :  ^1  funftionfi.i',  ij,  z.  ■  ■  •)  af  an//  nuiiihrr  of  rarldhlc^  As' 
rail  I'll  Jiniiio'jcneovs  if  the  fawtlnn  /.S  III  itltljtlu'il  hi/  SDliie  jinirer  I  if  X  ir/irii 

all  fJic  mrinhJi's  m-c  m nltipJlril  lij  X:   tind  the  power  of  \  which  factors 


--Q.-I 

(dS\ 
KdTjv 

'^•=a=^( 

'dS\ 

^.  =  ("1)  =r( 

t)' 

XdvJT          ^ 

\di-/T 

-Kill 

iv =-.(!'), 

\dvlr 

-■'-'(E- 

108  DIFFERENTIAL  CALCULUS 

out  is  called  the  order  of  homogeneity  of  the  function.  In  symbols  the 
condition  for  homogeneity  of  order  n  is 

f{Xx,Xi/,  Xz,  ...)  =  X\f(x,  y,  z,  . . .).  (29) 

Thus  xe^  +  '-^j       -^  +  tan-i-,  .  C29') 

X  z-  z  Va'-  +  / 

ai'c  liomogeneous  functions  of  order  1,  0,  —  1  respectively.  To  test  a 
function  for  homogeneity  it  is  merely  necessary  to  replace  all  the  vari- 
ables by  A  times  the  variables  and  see  if  A  factors  out  completely.  The 
homogeneity  may  usually  be  seen  without  the  test. 

If  the  identity  (29)  be  differentiated  with  respect  to  A,  with  x'=Xx,  etc., 

(^  a?  +  -^  ^7'  +  ■'  ^  +  ■  ■  ■)'^'^^'''  ^^'  ^•'-■'  ■  ■  •)  =  ^^^""'/(^'  yr^,--  •)• 

A  second  differentiation  with  respect  to  A  would  give 

Now  if  A  be  set  equal  to  1  in  these  equations,  then  cc'  =  cc  and 

x^~  %  +  2  xy  M-  +  //  %+2  xz  ,-.^  +  --.  =  n  (n  -  1  )/(,>■,  y,  z,  •  •  •)• 

In  words,  these  equations  state  that  tlic  sum  of  the  partial  derivatives 
each  multiplied  l)y  the  variable  with  I'cspect  to  Avhich  the  differentia- 
tion is  performed  is  n  times  the  function  if  the  function  is  homogeneous 
of  order  n ;  and  that  the  sum  of  tlic  second  derivatives  each  multiplied 
by  the  variables  involved  and  by  1  or  2,  according  as  the  variable  is 
repeated  or  not,  is  n  {n  —  1)  times  the  function.  The  general  formula 
ol)tained  by  differentiating  any  number  of  times  Avith  respect  to  A  may 
be  ex})ress('(l  symbolically  in  tlie  convenient  form 

(■'■/>..  +  il^Kj  +  .--/>.  +  •  ■  •)'/'=  ?'  ("  -  1 )  •  •  •  ('^  -  /■■  +  !)/•        (31) 
This  is  known  as  EiiJcr's  Fdniinl'i  on  homogeneous  functions. 

It  is  worth  while  nnlinii;  tliat  in  ;i  certiiin  sense  every  e(iuati()n  wliicli  represents 
a  .treonietric  <ir  pliysical  rehitinn  is  hoinoireneous.  Por  instance,  in  i^^eonietry  tlie 
magnitudes  tliat  arise  may  be  lenirths.  areas,  vohimes.  or  angles.  These  ma,<;ni- 
Uules  are  expressed  as  a  number  times  a  unit  ;  thus,  v2  ft.,  .3  sq.  yd.,  tt  cu.  ft. 


PARTIAL  DIFFERENTIATION;   EXPLICIT  109 

111  adding  and  subtracting,  the  terms  nuist  be  like  (piantities ;  lengtlis  added  to 
lengths,  areas  to  areas,  etc.  The  fundamental  unit  is  taken  as  length.  The  units  of 
area,  volume,  and  angle  are  derived  therefrom.  Thus  the  area  of  a  rectangle  or 
the  volume  of  a  rectangular  parallelepiped  is 

A  =  a  ft.  X  h  ft.  =  ab  f t.'-^  =  ah  sq  ft.,      V  =  a  f t.  x  b  ft.  x  c  ft.  =  abc  ft.^  =  abc  cu.  ft., 

and  the  units  sq.  ft.,  cu.  ft.  are  denoted  as  ft.-,  ft.^  just  as  if  the  simple  unit  ft. 
had  been  treated  as  a  literal  tjuantity  and  included  in  the  nudtiplication.  An  area 
or  volume  is  therefore  considered  as  a  compound  quantity  consisting  of  a  inimber 
which  gives  its  magnitude  and  a  unit  which  gives  its  quality  or  dimensions.  If  L 
denote  length  and  [i]  denote  "of  the  dimensions  of  length,"  and  if  .similar  nota- 
tions be  introduced  for  area  and  volume,  the  equations  [^4]  =  [L]-  and  [F]  =  [L]"' 
state  that  the  dimensions  of  area  are  squares  of  length,  and  of  volumes,  cubes  of 
lengths.  If  it  be  recalled  that  for  purposes  of  analysis  an  angle  is  mea.sured  by  the 
ratio  of  the  arc  subtended  to  the  radius  of  the  circle,  the  dimensions  of  angle  are 
seen  to  be  nil,  as  the  definition  involves  the  ratio  of  like  magnitudes  and  must 
therefore  be  n  pure  numJ)er. 

When  geometric  facts  are  represented  analytically,  cither  of  two  alternatives  is 
open  :  l'\  the  ecjuations  may  be  regarded  as  existing  between  mere  numbers  ;  or 
2°,  as  between  actual  magnitudes.  Sometimes  one  method  is  preferable,  sometimes 
the  other.  Thus  the  equation  x^  +  y'-^  =  r-  of  a  circle  may  be  interpreted  as  1°,  the 
sum  of  the  squares  of  the  coordinates  (numbers)  is  constant ;  or  2'^,  the  sum  of  the 
squares  on  the  legs  of  a  right  triangle  is  equal  to  the  square  on  the  hypotemise 
(Pythagorean  Theorem).  The  second  interpretation  better  sets  forth  the  true 
inwardness  of  the  equation.  Con.sider  in  like  manner  the  parabola  ?/-  =  ipx.  Gen- 
erally y  and  x  are  regarded  as  mere  numbers,  but  they  may  ('(jually  l)e  looked 
upon  as  lengths  and  then  the  statement  is  that  the  square  upon  the  ordinate  ecjuals 
the  rectangle  Tipon  the  abscissa  and  the  constant  length  4p  ;  this  may  be  inter- 
preted into  an  actual  construction  for  the  jiarabola,  because  a  sijuare  equivalent 
to  a.  rectangle  may  be  constructed. 

In  the  last  interpretation  the  constant  p  was  assigned  the  dimensions  of  length 
so  as  to  render  the  equation  homogeneous  in  dimensions,  with  each  term  of  the 
dimensions  of  area  or  [L]-.  It  will  be  recalled,  however,  that  in  the  delinition  of 
the  parabola,  the  quantity  p  actually  has  the  dimensions  of  length,  being  half  the 
distance  from  the  fixed  ]3oint  to  the  fixed  line  (focus  and  directrix).  This  is  merely 
another  corroboration  of  the  initial  statement  that  the  ecjuations  which  actually 
arise  in  considering  geometric  prolilems  are  homogeneous  in  their  dimen.sions,  and 
nntst  be  so  for  the  reason  that  in  stating  the  first  e(piation  like  magnitudes  nuist 
be  compared  with  like  magnitudes. 

The  question  of  dimensions  may  be  carried  along  through  such  processes  as 
differentiation  and  integration.  For  let  y  have  the  dimensions  [y]  and  x  the  dimen- 
.sions [x].  Then  Ay,  the  difference  of  two  ?/"s,  must  still  have  the  dimensions  [?/] 
and  Ax  the  dimen.sions  [x].  The  quotient  Ay/ Ax  then  has  the  dimensions  [2/]/[x]. 
For  example  the  relations  for  area  and  for  volume  of  revolution. 


.,        .         [dAl      [.I]  [dVl      [V]       ^^^. 


dx       '  '     (7x 


and  the  dimensions  of  the  left-hand  side  check  with  those  of  the  right-hand  side. 
As  integration  is  the  limit  of  a  sum,  the  dimensions  of  an  integral  are  the  product 


110  DIFFERENTIAL  CALCl'LUS 

of  the  dimensions  of  the  function  to  be  inte.irrated   and  of  the  differential  dx. 
Thus  if 


r  "^     dx  1  ,  X 

=  I     =:  -  tan-i  -  +  c 

Jo  a-  +  X-      a  a 


y     .     .,^ 

were  an  integral  arising  in  actual  practice,  the  very  fact  that  a'^  and  x'^  are  added 
would  show  that  they  nuist  have  the  same  dimensions.  If  the  dimensions  of  x 
be  [i],  then 

and  this  checks  with  tlie  dimensions  on  the  right  which  are  [i]~S  since  angle  has 
no  dimeiisions.  As  a  rule,  the  theory  of  dimensions  is  neglected  in  x^ire  mathe- 
matics :  l)ut  it  can  nevertheless  be  made  excei'ilingh'  useful  and  instructive. 

In  mechanics  the  fund(i)Hcnt<il  iniit^  are  length,  mass,  and  time  ;  and  are  denoted 
by  [i].  [-V].  [7'].    The  fullowing  table  contains  some  derived  units: 

[LI  ,     .     [L]  ^  r-virLi 

velocity  - —  ,         acceleration- — -,  force  — -    —^ , 

,     ,    .        [ii-       ,     .  [M]  [-virLi 

areal  velocitv       ^ — ~,       density  - — -,  momentum' — —^, 

^         [T]  [L]3  IT] 

,  .     1  r-vuLi-  i^nm- 

anyular  velocity ,        moment  - — ^-^ — —,       cnerijy  '■ — —^—^. 

-   [T]  [T]-^  '  [TJ^ 

With  the  aid  of  a  table  like  this  it  is  easy  to  convert  magnitudes  in  one  set  of 
units  as  ft.,  lb.,  sec,  to  another  system,  .'^ay  cm.,  gm..  sec.  All  tliat  is  necessary  is 
to  substitute  for  each  individual  unit  its  value  in  the  new  system.    Thus 

q  =  321  ^^  ,        1  ft.  =  30.48  cm..         ,/  ^  32'  x  30.48  — "'   =  080]-  -^^^. 
sec-  '  sec-  '  sec- 


EXERCISES 

1.  Obtain  the  derivatives/",..  /,',',.  /,J^,,  J]'/,^  and  verify/,"^  =/,',',■■ 

{a)  sin-i  •' ,  (/J)   Ing  -^^i^^' ,  [y)  0 1'i]  +  ^  (,r//). 

X  xy  \xl 

2.  Compute  c-v/ci/-  in  polar  cuilrdinates  by  the  straightfm-ward  method. 

3.  Show  that  '(-  — -  =  '-'-  if  v  =/(x  +  at)  +  4>{x  —  (d). 

cx-       ct'^ 

4.  Show  that  tliis  eijuation  is  unchanged  in  form  Ijy  the  transformation  : 

;;^^  +  2  x//-  ,■  -  +  2  (//  -  ?/•'■)  _■-  +  x-y-f  =  0  ;       u  =  xy.     v  =  1/y. 
cx-  (X  cy 

5.  In  polar  coordinates  z  =  r  cosi9.  x  —  r  sin  ft  cos(^,  //  -=  r  sin  ft  sin  (f>  in  space 

fX-       (y-       (y-       i-l(r\      cr!       s\\\-ft((f>-       >h\ft(ft\  (ft'\ 

The  work  of  transformation  may  be  shortened  by  substituting  successively 
X  =  ;'j  C()S  (/>.     //  =  r,  sin  </j.      and      ,~  =  /•  cos  0.     z',  —  ;•  sin  0. 

6.  T>et  X.  )/.  z.  t  be  four  imlejiendent   \;!ria1iles  and  ./•  —  reos«/i.  y  —  )-sin0,  z  - 
tlie  e(|nations  for  transfoniiing  ,/•.  y.  x  to  eylindriral  eorniiinaies.    Let 


PAirriAL  DIFFERENTIATION;   EXPJ.KIT  111 

cxcz  c!/cz  C.C-       cij-  cycl  cxct 

show     Z  = ,     X  cos  0  +  i  sill  0  = ,     V  sill  0  —  (r  cos  0  = ^ , 

/•  cr  r  cz  r  ct 

where  ;— 'V  =  U/(>'-    (' 'i"  iniimrtance  fnr  the  Hertz  osciUator.)      Take  cf  /  c(p  =  0. 

7.  ^Vpply  the  test  fur  an  exact  differential  t(»  eacii  of  the  fnllowjng.  and  write 
by  inspectinn  the  functions  correspondini;-  to  the  exact  differentials: 

{a)  Sxdx  +  y-dy,         (p)  Zxydx  +  x^dy.         (7)  x-ydx  +  y-dy, 
1 5 ,  xdx  +  ydy  xdx  -  ydy  ydx  -  xdy 

X-  +  y-  X'  +  y^  X-  +  y- 

{■n)  (4  x"  +  8  x-y  +  r )  f/x  +  (x-^  +  2  xy  +  3  ;/■■')  %.         {9)  x-y-  {dx  +  dy). 

8.  Express  the  conditions  that  P{x.  y.  z)dx  +  ^^M.'".  y.  z)dy  +  R(x.  y.  z)dz  be 
an  exact  differential  dF{x.  y.  z).    Apph'  these  Cdiiditions  to  the  differentials  : 

(a)  3  x-y-zdx  +  2  xhjzdy  +  x^y-dz,         (/i)  {y  +  z)  dx  +  {x  +  z)  dy  +  (x  +  y)  dz. 

9.  (>btaiu  (- — I  =  (  — I  and  (  —  1  =  I- —    frniu  (•17)  with  proper  variables. 

UrJ,.     Uv/t        WN/;>      vO>^ 

10.  If  three  functions  (called  thermodynamic  potentials)  l)e  defined  as 

■^  =  r-Ts.  x  =  i'  +  p^--         ^^r-rs  +  pv^ 

show  df  =  —  Sd  T  —  pdf,     dx  =  Tds  +  vdp.     r/j-  =  —  Sd  T  +  vdp, 

and  express  the  conditions  that  df.  dx.  d^  be  exact.    Compare  with  Ex.  0. 

11.  State  in  words  the  definitions  corresponding  to  the  defining  formulas,  p.  107. 

12.  If  the  sum  (Mdx  +  Xdy)  +  {Pdx  +  (^dy)  of  two  differentials  is  exact  and  one 
of  the  differentials  is  exact,  the  other  is.    Prove  this. 

13.  Apply  Euler"s  Formula  (31),  for  the  simple  case  k  =  1,  to  the  three  func- 
tions (29')  and  verify  the  formula.    Apply  it  for  k  =  2  to  the  lirst  function. 

14.  A'erify  the  hoiiiogeiieit}''  of  these  functions  and  determine  their  order  : 
(a)  y-/x  +  X  (log X  -  log  y),         {(3)       ''■'"■'"      ,  (7)  ■'^^~ 


AX-  +  y-  ^«  +  f^y  +  cz 

( 5 )  xy^  +  z\.  {e)sG.  cot- 1  ^^  ( n  ^r--^^ ' 

z  VX  -f   V  y 

15.  State  the  dimensions  of  moment  <if  inertia  ami  convert  a  unit  of  moinent  of 
inei'tia  in  ft. -Hi.  into  its  equivalent  in  cin.-gm. 

16.  Diseuss  for  dimensions  Peirce's  formulas  Nos.  03.  124-125.  220.  .300. 

17.  (  out  1  line  Ex.  1  (.  p.  101.  to  show      -  -  =  —  ami =  inc 1 

dt  c'li       a/i  dt  ci'n  cq;       oj, 

18.  If  /',-  =  —  in  Ex.  17.  p.  101.  show  without  analvsis  that  2T  =  7,;^,  -f-  '}.,/'.,. 

If  T'  denote  T'  =  7'.  wliere  7"  is  consiilcred  as  a  function  of  ji^.  p.,  while  T  is  con- 
sidered as  a  function  of  q^.  I'j.,.  prove  from  7"  =  'JiPi  +  <'l-^P->  —  "^  fli'it 

f  r  _  .        '^L-  _IL 

cpi  cq,  tqi 


112  DIFFERENTIAL  CALCULUS 

19.  If  (Xj,  ?/,)  and  (x,^,  t/.,)  are  the  coordinates  of  two  moving  particles  and 

d'^x,       _,,  d-y,       ,^  d-x.,       ^,  d-y„ 

'  df^  '         ^  (Zi2  ^'        -  dfi  ^        -  di2  2 

are  the  equations  of  inoti<^n,  and  if  Xj,  y^,  x.,,  y^  are  expressible  as 

in  terms  of  three  independent  variables  r/^,  q.-,,  q,^,  show  that 

fT/j  Cf/i  cq^  cq^        dtcq^        cq^ 

where  T  =  I  [m^v^  +  m.^v.^)  =  T{q^,  q„,  q^,  q^,  q.,,  q„)  and  is  homogeneous  of  the 
second  degree  in  r/j,  7.,,  f/3.  The  work  may  be  carried  on  as  a  generalization  of 
Ex.  17,  p.  101,  and  Ivx.  17  above.  It  may  be  further  extended  to  any  lunnber  of 
particles  whose  positions  in  space  depend  on  a  number  of  variables  q. 

20.  In  Ex.  19  if  Pi  =  — ,  generalize  Ex.  18  to  obtain 

cqi 

'ii  =  —  '    —  =  ---'    Q^=  -^  +  — • 

cPi         cqi  cqi  dt         cq^ 

d  cT       cT        ,   ,        dpi      cT  .     ,     .,      -r 

The  equations  Q,-  = and  %  =  -^ — | are  respectively  the  Lagran- 

dt  a'li       an  dt        cqi 

gian  and  Hainiltonian  ecpiations  of  motion. 

21.  If  rr'  —  k-  and  (p'  =  (p  andf'(>'',  0')  =  v{r,  0),  show 

c~v'       ]  cv'        1    c-v'       f^  /cH       1  cv       1   (-V 

T^  +  "--;  +  ^  ^T7  =  ^  ^  +  "  ^  +  ^  -   :, 
cr  -      r  cr       r  -  ccp  -       r-  \cr-       ?•  cr      r-  c  (p- 

22.  If  /t'  =  k~,  (p'  —  (p,  6'  =  6,   and   i-'(K.  4,'.  0')  —  -  v{i\  <p,  6),  show  that  the 

/■' 

expression  of  Ex.  o  in  the  primed  letters  is  kr-/r''''  of  its  value  for  the  uiiprimed 
letters.    (Useful  in  §  1U8.) 

23.  Uz'-^xJ"\^i{'i\,  shew  x-5'^+2x///^+;/2f^  =  0. 


cxcy  cy- 

24.   Make  the  indicated  cliani,^es  of  varialile  : 


,    .    c"V       c"V  „    /('-V       c"V\  .. 

(a) f-      —  =  c-  - " I \ )  II  X  =  e"  cos  v.  y  =  e"  sm  v, 

cx~        cy~  \cu-        cv- / 

cu-        cv-       \cx-        cy-/l\cu/       \ft7J 

F  T  r  rh  7  f  rrfi 

x=f(u,v),     y  =  <p{i(,v), 


cu       cv       cv  cu 

25.  For  an  orthogonal  transf(n-ination  (Ex.  10  (f),  y).  100) 

c-v       r-v       (-V       c-v        f'-r        (-V 
r--,  +  .    ,+  .    ,=  .   7;+       ;;  +  — ;7- 
ex-       ( y-       (Z-       cx  -       ((/-       (Z- 

54.  Taylor's  Formula  and  applications.   The  dcvclojnnoiit  ()l\f(.<\  //) 
is  found,  as  was  tlu'  Tli('()r(Mn  of  tlu.'  Mean,  from  the  relation  (p.  Uoj 


PARTIAL  DIFFEKENTIATIOX;   EXPLICIT  113 

A/  =  $  (1)  -  4>  (0)      if     $  (t)  =  f(a  +  f/i,  h  +  tk). 
If  ^(t)  be  expanded  hy  ^Maclaiiriir's  Formula  to  n  terms, 

The  ex})ressions  for  <J>'(/)  and  $'(0)  may  be  found  as  folloAvs  by  (10)  : 

<^'(0  =  A/::  +  ¥;„  *'(0)  =  [a/::  +  ^:/;;]-- 

then  ci,"(;^)  =  /^  (/,/;;;  +  /.q  +  a- (///:;  +  /;C) 

^'('XO  =  {hir,.  +  /^ /),)'>;    <f^'>(0)  =  lihD,.  +  /.■/),)'■/]._„. 

And  f(,,  +  //,  //  +  /,•)  -/(r/,  //)  =  A/=  $(1)  -$(0)  =  (A/)„  +  l-D,),f\n,  h) 

+  ^  (/' A„.  +  /.-AjyC",  /')  +  •  •  •  +  ^;^3iyi  i^'^^'-  +  /•■^^.)"~V'(",  ^0 

+  ^  {hD^,  +  h-D,;)"f{"  +  ^//,  /'  +  ^A-).  (32) 

In  this  expansion,  the  increments  A  and  /.•  may  be  replaced,  if  de- 
sired, by  X  —  a  and  //  —  !>  and  then  /(,/■,  //)  will  be  expressed  in  terms 
of  its  value  and  the  values  of  its  derivatives  at  (a,  h)  in  a  manner 
entirely  analogous  to  the  case  of  a  single  variable.  In  particular  if  the 
point  (ir,  h)  about  Avhich  the  development  takes  place  be  (0,  0)  the 
development  becomes  ]Maclaurin's  Formula  for  /(.r,  //). 

f(^;  V)  =/(o,  0)  +  {..■D,  +  z/A,)/\o,  0)  +  ^  (,.i;,  +  >iii,ff{0, 0)  + . . . 

+  ^^i-^.,C'' Z^.,. +  ///>,)"  "^(0,  0)  +  ^  {.rir.  +  yD,y'f{6.r,  On).      (32') 

Whether  in  j\[aclaurin"s  or  Taylor's  I'^oi'nuda,  the  successive  terms  are 
homogeneous  polynomials  of  the  1st,  2d,  •  •  ■,  (/?-  —  l)st  order  in  ,/■,  y  or 
in  X  —  n,  y  —  h.    The  formulas  are  uni(]ue  as  in  §  32. 


Suppose  V  1  —  x-  —  ij~  is  to  be  developed  about  (0,  0).    The  successive  deriva- 
tives are 

./:;  =  ~7=^^='  /;  =  -^^^^=.  .ceo,  o)  =  o,  /,;(o.  n)  =  o, 

Vl  —  X-  —  y~  A  1  —  X-  —  ;/- 

•f  .r.r  ,  _     s  '       ■' .1-1/  _  X  '       •''/ 


,„  _     \(\  —  //-)x  ,„  _  ?/■'  -  2  ,c//-  -  y 

{\-x?--y-'-Y  ■        (1 -.C- -//-)- 


and     Vl  -  .f-  -  y-  =  1  -h  (0 .r  +  0 //)  +  ',(-  •*■-  +  0 xy  -  ;/-)  +  ;i  (0 ,/•■•  +  ...)  +  ..., 
"I"        V  1  —  X-  —  ;/-  =  1  —  ;■,  (x-  -1-  ?/-)  -f-  terius  of  fourth  order  -(-•••. 

In  tiiis  case  the  expansion  may  be  found  by  trcatinii;  x-  -|-  //-  as  a  sinule  term  and 
expanding  by  the  binomial  theorem.    The  result  would  be 


114  DIFFEKEXTIAL  CALCULUS 

[1  -  {X-  +  7/2)]  i  ^l-l(x-  +  r)  -  I  {x^  +  2, /:-//-  +  >/)  -  ^\  {X-  +  i/'f . 

That  the  development  thus  obtaineil  is  iilentical  witli  the  Maehiiirin  developineiit 
tliat  might  be  had  by  the  method  above,  follows  from  the  uni(iueiiess  of  the  devel- 
opment.   Some  such  short  cut  is  usually  available. 

55.  The  condition  that  a  function  ;-;  =  /'(,/■,  //)  have  a  niininnnii  or 
niaxiuiuni  at  (",  f>)  is  tliat  A/'>  0  or  A/'<  0  for  all  values  of  //  =  A,/' 
and  k  =  A//  Avhicli  are  sufticiently  small.  From  cither  geometrical  or 
analytic  considerations  it  is  seen  that  if  the  surface  ;■  =f(x,  jf)  lias  a 
minimum  or  maximum  at  (n,  />).  the  curves  in  which  the  ])lanes  //  =  // 
and  .r  =  a  cut  the  surface  have  minima  or  nuixima  at  ,/■  =  >i  and  //  =  /> 
respectively.  Hence  the  partial  derivatives/','  and  /'J  must  l)oth  vanisli 
at  ((I,  li),  provided,  of  course,  that  exceptions  like  those  mentioned  on 
page  7  be  made.    The  two  simultaneous  e(|uations 

f:  =  Q,  /:;  =  o,  (:w) 

corresponding  to /''(,'•)  =  0  in  the  case  of  a  function  of  a.  single  varia- 
ble, may  then  be  solved  to  iind  the  })()sitions  (./',  //)  of  the  minima 
and  maxima.  Frequently  the  geometric  or  ]»hysical  intcrpi'etatioii  of 
,'.■  =_/'(',/■,  //)  or  some  special  device  will  then  determine  whether  there 
is  a  maximum  or  a  minimum  or  neither  at  each  of  these  ])oints. 

For  example  let  it  be  required  tn  find  the  niaxinium  rectangular  parallelepiped 
which  has  three  faces  in  the  cdrirdinate  planes  and  one  vertex  in  the  plane 
x/a  -f  y/h  -\-  z/c  —  1.    The  volume  is 


y  =  xiiz  =  cxii  1 1  — 

\      II     I. 

cV  c  r,  fT  r  r 

--~=  —  2'-  XII  —  -  If-  +  f//  =  0  -  =  —  2  -  xy x-  -1-  ex  =  0. 

ex  a  b  ly  b  n 

Tlie  solution  of  tliese  eijuations  is  x  =  I  n.  y  =  1  b.  The  corresp(iiiding  z  is  '.  /■  and 
the  volume  U  is  therefore  (ibr/U  or  -,  of  the  volume  cut  off  from  the  lirst  octant  by 
the  i)lane.  It  is  evident  that  this  solution  is  a  maximum.  There  are  other  solutions 
of  I'j'  =  Uj  =  0  which  have  been  discarded  because  tliej'  give  V  =  0. 

The  conditions/','  =/'„'  =  0  nitty  be  esttiblishcd  tmtdytically.    For 

A/=(/:;  +  ^,)A,''  +  (/;  +  t,)A//. 

Now  ;is  ^j,  ^,  are  infinitesimals,  the  signs  of  tlio  ]i;ircnthcscs  ;irc  deter- 
mined ly  the  signs  of/',',/j,'  unless  these  derivatives  vanish;  ;ind  hence 
unless/','  =  0,  the  sign  of  A/'  for  A.''  sutfieiently  small  and  ]M)siti\-e  and 
A// =  0  would  be  opposite  to  the  sign  of  Af' for  A.'' sutfieiently  small  ;nid 
negative  and  A// =  0.  Tlierefore /'//•  -■/  in'nn  nm  m  nr  ni'i.ri  m  n  m  j][  =  0  : 
(1)1(1  in  ///v;  iiKiiiiicr  _/'^' =  0.  Considerations  like  these  will  ser\'e  to 
esttiblish  a  criteri(Mi  iV)r  distiiio'tiishiiig'   lietwecn   maxima  tiiid   niininia 


PARTIAL  DIFFERENTIATION;   EXPLICIT  115 

analogous  to  the  criterion  furnished  by  /"(.'•)  in  the  case  of  one  vari- 
able.   For  if /j'  =f'^  =  0,  then 

by  Taylor's  Formula  to  two  terms.  Xow  if  the  second  derivatives  are 
continuous  functions  of  (.'■,  //)  in  the  neighborhood  of  {<i,  h),  each  deriv- 
ative at  {((  -\-  6Ji,  h  +  61: )  nray  be  written  as  its  value  at  (ji,  h)  plus  an 
infinitesimal.    Hence 

Now  the  sign  of  \f  for  sufficiently  small  values  of  //,  /.•  must  bo  the 
same  as  the  sign  of  the  first  jjarenthesis  provided  that  parenthesis  does 
not  vanish.    Hence  if  the  quantity 

.,  ,  ,,/       ,.  .  -,   .„       ,1  ./,  >  0  for  everv  ('/^  /.•),  ^  minimum 

^  "         ^  Ji(  '  >  ^  0  for  every  (A,  /.),  a  maximum. 

As  the  derivatives  are  taken  at  the  point  {a,  b),  they  have  certain  constant 
vahies,  say  ^1,  B,  C.  The  qnestion  of  distinguishing  between  minima  and  maxima 
tlierefore  reduces  to  the  discussion  of  the  possible  signs  of  a  quadndk  fdnii 
Ah- +  2  Bhk  +  Ck-  for  different  values  of  h  and  k.    The  examples 

k-2  +  f-^,    -  /(^  -  k^,     /(2  _  A-^    ±  {h  -  A-)--^ 

show  that  a  (piadratic  form  may  be  :  eitlier  P,  positive  fur  every  {//.  k)  except  (0,  0) ; 
or  2^,  negative  for  every  (li.  k)  except  (0.  0);  or  ,S^,  i)o.sitive  for  sume  values  (/(,  k) 
and  negative  for  otliers  and  zero  for  others  ;  or  linally  4^,  zero  for  values  other  than 
(0.  0),  but  either  never  negative  or  never  positive.  Moreover,  the  f(jur  possiljilities 
here  mentioned  are  the  only  cases  conceivable  except  .j\  that  A  =  B  —  C  =  0  and 
the  form  always  is  0.  In  the  first  case  the  form  is  called  a  definite  positke  form,  in 
the  .second  a  definite  negidive  form,  in  the  third  an  indefinite  form,  and  in  the  fourth 
and  fifth  a  singular  form.  The  tirst  case  asstires  a  mininmm,  the  .second  a  maxi- 
mum, the  third  neither  a  minimum  nor  a  maxinmm  (sometimes  called  a  minimax)  ; 
but  the  case  of  a  singular  form  leaves  the  question  entirely  undecided  just  as  the 
condition /"{/)  =  0  did. 

The  conditions  which  distinguish  between  the  different  possibilities  may  be  ex- 
pressed in  terms  of  tlie  coefficients  .1,  7>,  C. 

I'pos.  def.,     ;;■-'<. IC.     A.OQ;         S'indef.,     B- >  AC  ■ 
2^neg.  def.,     11- <  AC.     A.C<0;         4^  sing.,       B- =  AC. 

The  conditions  for  distinguishing  between  maxima  and  minima  are  : 
/",  =  0  "1         „  ,        r  /■''..  /"'  >  0  minimum  ; 

./,/  =  '-^  J  \fr,-J,i.i  ^  "^  maxnuum  ; 

It  may  be  noted  that  in  applying  these  conditions  to  the  case  of  a  definite  form  it 
is  sufficient  to  show  that  ei 
sarilv  base  the  same  sign. 


116  DIFFERENTIAL  CALCULUS 

EXERCISES 

1.  "Write  at  length,  without  syiiil)olic  shortening,  the  expansion  of /(x,  ?/)  by 
Taylor's  Formula  to  and  including  the  terms  of  the  third  order  in  x  —  a,  y  —  b. 
"Write  the  formula  also  with  the  terms  of  the  tliird  order  as  the  remainder. 

2.  "Write  by  analogy  the  proper  form  of  Taylor's  Formula  for/(x,  y.  z)  and 
prove  it.    Indicate  the  result  for  any  mimber  of  variables. 

3.  Obtain  the  quadratic  and  lower  terms  in  the  development 

(rt)  of  xy-  +  sin  xij  at  (1,  I  ir)     and     (/3)  of  tan-i  {y/x)  at  (1.  1). 

4.  A  rectangular  parallelepiped  with  one  vertex  at  the  origin  and  three  faces 
in  tile  coordinate  planes  has  the  opposite  vertex  upon  the  ellipsoid 

^-/"- +  yV'^- +  --/'•- =1. 

Find  the  maximum  volume. 

5.  Find  the  point  within  a  triangle  such  that«the  sum  of  the  squares  of  its 
distances  to  tlie  vertices  shall  be  a  mininmm.  Note  that  tlie  point  is  the  intersec- 
tion of  the  medians.    Is  it  obvious  that  a  mininmm  and  not  a  maxinunn  is  present  ? 

6.  A  floating  anchorage  is  to  be  made  with  a  cylindrical  body  and  equal  coni- 
cal ends.    Find  the  dimensions  tliat  make  the  surface  least  for  a  given  volume. 

7.  A  cylindrical  tent  has  a  conical  roof.    Find  the  best  dimensions. 

8.  Apph"  the  test  Ijy  second  derivatives  to  the  problem  in  the  text  and  to  any 
of  Exs.  4-7.    Discuss  for  maxima  or  minima  the  following  functions  : 

{a)  x-y  +  xy-  —  x,  (/3)  x"  +  y^  -  i'-^-  -  \  (^-  +  y-), 

(7)  X-  +  y-  +  X  +  y,  (5)   J  y-  -  xy-  +  x-y  -  x, 

( e )  x^  +  y-'  -  0  xy  +  21,  ( j")  .'•"'  -\-  y^  -  2  x-  +  Axy-±  y^. 

9.  State  the  conditions  on  the  first  derivatives  for  a  maxinmm  or  minimum  of 
function  of  three  or  any  luunber  of  variables.    Prove  in  tlie  case  of  three  variables. 

10.  \  wall  tent  with  rectangular  Ijody  and  gal>le  roof  is  to  be  so  constructerl  as 
to  use  tlie  least  amount  of  tenting  for  a  given  volume.    Find  tlie  dimensions. 

11.  Given  anj'  number  of  masses  m^.  in.,.  ■  ■  ■.  m,,  situated  at  (.Cj,  y^).  (x.,.  ?/.,),  •  ■  •, 
(•C«.  y?,)-  Show  that  the  point  about  whicli  their  moment  of  inertia  is  least  is  their 
center  of  gravity.  If  the  points  were  [x^,  //j,  2j),  •  •  ■  in  si)ace,  what  point  would 
make  ^inr-  a  minimum  '.' 

12.  A  test  for  maximum  or  minimum  analogous  to  that  of  Ex.  27,  }).  10,  maj' 
be  given  for  a  function /(j,-.  y)  of  two  varialiles,  namely  :  If  a  function  is  positive 
all  over  a  region  and  vanishes  upon  tlie  contour  of  the  region,  it  must  have  a  max- 
inmm within  tlie  region  at  tlie  jxiint  for  wliich  ./'_'  =./'„'  =  0.  If  a  function  is  Unite 
all  over  a  region  and  becomes  intinite  over  tlie  contour  of  tlie  region,  it  must  have 
a  minimum  within  the  region  at  the  point  for  which /^  =/J  =  0.  These  tests  are 
subject  to  the  proviso  tliat/,'  =,/'J  =  0  has  only  a  single  solution.  Comment  on  the 
test  and  ai)ply  it  to  exercises  above. 

13.  If  '(.  I).  I',  r  arc  tiic  sides  of  a  given  triangle  and  the  radius  of  tlie  inscrilied 
circle,  tlie  i>yraiiiid  of  altitude  //  constructed  on  the  triani;ie  as  base  will  have  its 
maximum  surface  wlieii  the  surfaci'  is  ',  (a  +  6  +  cj^'r-  +  h'-. 


CHAPTER  V 


PARTIAL  DIFFERENTIATION;    IMPLICIT  FUNCTIONS 

56.  The  simplest  case  ;  F(Xf  y)  =  0.    The  total  differential 
dF=  F[jl.r  +  F/h/  =  dO  =  0 


indicates 


dx 


F' 


'/// 


f: 


(1) 


F(x,y)=0 


as  the  derivative  of  y  by  a*,  or  of  x  by  y,  where  y  is  defined  as  a  function 
of  .r,  or  X  as  a  function  of  y,  by  the  relation  F{.r,  y/)  =  0  ;  and  this  method 
of  obtaining  a  derivative  of  an  imj^liclt  function  without  solving  expli- 
citly for  the  function  has  probably  been  familiar  long  before  the  notion 
of  a  partial  derivative  was  obtained.  The  relation  F(x,  y)  =  0  is  pictured 
as  a  curve,  and  the  function  y  =  4*(x),  whicli  would  be  obtained  by  solu- 
tion, is  considered  as  multiple  valued  or  as  restricted  to  some  definite 
Ijortion  or  branch  of  the  curve  F(x,  y)  =  0.  If  the  results  (1)  are  to 
be  applied  to  find  the  derivative  at  some  i)oint 
(x^^,  yj  of  the  curve  F(x,  y)  =  0,  it  is  necessary 
that  at  that  point  the  denominator  F^  or  Fj  sliould 
not  vanish. 

These  i)ictorial  and  somewliat  vague  notions 
may  be  stated  precisely  as  a  tlicorcm  susce})til)le 
of  proof,  namely  :  Let  ./■  be  any  real  value  ui  x 
such  that  1°,  the  equation  F{x^^,  v)  =  ^  l^'i^  ^  i'*'*'!  solution  y^ ;  and  2°,  the 
function  F{.i-,  y)  regarded  as  a  function  of  two  independent  variables 
(./',  ?/)  is  continuous  and  has  C(jntinuous  first  })artial  derivatives  F,',  F',^  in 
the  neighborhood  of  {x^^,  y^y^  and  ?°,  \\w.  derivative  F^'C'Vu  .Vo)  ^  ^  <\-<d^^ 
not  vanish  for  (.z'^,  y^) ;  tlien  F(./',  //)  =  0  may  be  solved  (theoretically) 
as  y=cf)(x')  in  the  vicinity  of  x  =  x^^  and  in  such  a  manner  that 
y .  =  <^ (.'•-),  that  (f>(x)  is  continuous  in  ./■,  and  that  cji(x~)  has  a  derivative 
(f>'(x)  =  —  F',./F'^  ;  and  tlie  solution  is  unique.  This  is  the  fundamental 
theorem  on  implicit  functions  for  the  sinq)le  case,  and  the  ])roof  follows. 

By  the  conditions  on  F^',  F'    tlie  Theorem  of  the  Mean  is  applicable.    Hence 

F  {X.  y)-F  (/„ .  2/„)  =  F{x,y)  =  (h  F^  +  kP;),^  +  ,,,  ,,„ ..  g,.  (2) 

Furthermore,  in   any  S(iuare  \h\<8,  |A-]<5  surroundimr  (./•„.?/„)    and   sufficiently 


small,  the  continuity  of  /■"'  insure: 


[F,'|<3fan(l  the  continuity  of  Fj  taken  with 
117 


118 


DIFFEKEXTIAL   (^VLCULUS 


1' 

X 

1 

1 

5 

1 
1 

/ 

/ 

6 

1 

.%-5 

1 
1 

1 
1 
1 

O 

25 

m 

A' 

the  fact  tliat  F^j{x^,,  2/^)  t^  0  insures  |Fj|>»i.    Consider  the  riiji.i^e  of  x  as  furthei 

restricted  to  values  sucli  that  | x  —  x^  |  <  ?/i5/3/  if  )n<M.    Now  consider  tlie  valuM 

of  F(x,  y)  for  any  x  in  tlie  permissible  interval 

and  for  y  =  y^  +  S  or  y  =  y^  —  d.    As  j kF^ \>iiid 

but  |(x  —  Xy)F^.|<7H5,    it   follows   from   (2)  that 

F{x,  Vq  +  5)  has  the  si,!i;n  of  5F^  and  F{x,  y^^  —  8) 

has  the  sign  of  —  5F^  ;  and  as  the  sign  of  F^^  does 

not  change,  F{x,  y^^  +  5)  and  F{x,  2/^  —  5)  have 

opposite  signs.    Hence  by  Ex.  10,  p.  45,  there  is 

one  and  oidy  one  value  of  y  between  2/y  —  5  and 

?/„  +  5  such  that  F{x,  y)  —  0.    Tlius  for  each  x  in 

tile  interval   there  is  one   and   only  one   y  such 

tliat  F{x,  y)  =  0.   The  eijuation  F(x,  ?/)  =  0  has  a 

unique  solution  near  (.r,„  ?/,,).    Let  y  —  (p(x)  denote  tiie  solution.    The  solution  is 

continuous  at  x  =  Xy  because  \y  —  y^^\<S.    If  (x,  y)  are  restricted  to  values  y  =  <^< (x) 

su<'h  that  F(x,  y)  =  0,  etjuation  (2)  gives  at  once 

k  _y-  ?/n  _  A^/  _  _  K{-^  +  Gh,  y  +  6^-^  d;/  _       F'.  (x,-,.  ?/„) 

it  ~  X  -  X 0  ~~^x~      F'^{x^  Oh.  y  +  'eJ) '        dx'"  ^(x„.  .vj  ■ 

As  7^^,,  F,^  are  continuous  and  F,^  ^  0,  tlie  fraction  k/]i  approaches  a  limit  and  the 
derivative  <^'(Xy)  exists  and  is  given  l)y  (1).  The  same  reasoning  would  apply  to 
any  point  x  in  the  interval.  Thi'  theorem  is  completely  pro\cd.  It  may  be  added 
that  the  expression  for  4>'{x)  is  such  as  to  show  that  0'(x)  itself  is  continuous. 

The  valttes  of  liiglicr  derivatives  of  iin])licit  functions  are  obtainable 
l»v  sticeessive  total  differentiation  as 


K  +  f;>/'  =  0, 
f;.:  +  2  7-::;//'  +  fZ;/'  +  f;,;/'  =  0, 


(3) 


etc.  It  is  notewortliy  that  tliese  successive  eijuations  may  he  solved  for 
tlie  derivative  of  liiyhest  order  hy  dividin;^-  ]>y  7"J  which  luis  heeu  assumed 
not  t(j  vanish.  Tlie  question  of  vvlicther  the  function  //  =  <^(..'')  defined 
iuqdicitly  hy  Fi'.r^  if)  =  0  has  deri\-atives  of  order  hi,L;her  tlian  tlie  tirst 
may  l)e  seen  hy  these  e(|uations  to  de])end  on  whether  P^ix,  //)  has 
liiLi'lun'  ])artial  derivatives  which  are  continuous  in  (./•,  //). 

57.  To  hnd  the  hkli-'hiki  (nul  iiiiniiiKi  of  y  =  ^  i^.r).  tliat  is,  to  find  the 
|ioints  wliere  the  tan,L;-ent  to  F(.'',  //)  =  0  is  jiarallel  to  the  .c-axis,  uliserve 
that  at  such  jioints  //' =  0.    Etjuations  (;>)  gi\'e 


F 


0, 


f;:...  +  Fy  =  0. 


(1) 


Hence  always  under  tlie  assTim];)tion  that  Fj  -^  0.  fliri'c  ari-  unixhiiii  ni 

fill'    infrrsrrflniis   of  F  =  0    "/."/    F'  =  0    if  1-"/^     (IU<I    ]f^  Imrr   fin'  snuir  s'i'jn, 

'tiiil  iitin'niKi  itf  fill-  iiifrrsrcfidiis  fir  irh'icli  /•','.  II  ml  F[^  Imrr  njijuisltt:  til'jns  ; 
the  case  F''.~  0  still  remains  undecided. 


PARTIAL  DIFFERENTIATIOX  ;   IMPLICIT  119 

For  example  if  F{x.  y)  =  x^  +  y^  —  Saxy  =  0,  the  derivatives  are 

dij  x~  —  ay 


3  (x2  -  ay)  +  3  {y'^  -  ax)  y'  =  0, 
6x-6ay'  +6 yy''^  +  3  (y^  -  ax) y"  =  0, 


dx  y-  —  ax 

d-y  2  a'^xy 


dx-  {y~  —  ax)'^ 

To  find  the  maxima  or  minima  of  ?/  as  a  function  of  x,  solve 

F;  =  0  =  j-  -  «?/,         F  =  0  =  x3  +  ?/3  -  3  axy,         F',^  yt  0. 

The  real  solutions  of  F;  =  0  and  F=  0  are  (0,  0)  and  (j^2a.  J/4«)  of  which  the 
first  must  be  discarded  because  F^'(0,  0)  =  0.  At  (V2«,  v-4a)  tlie  derivatives 
F,^  and  F,''.  are  positive  ;  and  the  point  is  a  maximum.  The  curve  F=  0  is  the 
folium  of  Descartes. 

The  role  of  the  variables  a-  and  y  may  l)e  interchanged  if  F^.  4^  0  and 
the  equation  F{j',  //)  =  0  may  be  solved  for  ./•  =  ^i'.i)-,  the  functions  ^ 
and  \\i  being  inverse.  In  this  way  the  vertical  tangents  to  the  cur\'e 
F  =  0  may  Ije  discussed.  For  the  ])oiiits  of  F  =  0  at  which  botli  F,'  =  0 
and  Fj  =  0,  the  equation  cannot  be  solved  in  the  sense  here  defined. 
Such  points  are  called  sinrjuhir  polnfa  of  the  curve.  Tlie  (questions  of 
the  singular  points  of  F  =  0  and  of  maxima,  minima,  or  minimax  (§  o7)) 
of  the  surface  .-.-  =  -?"(•'',  y)  are  related.  For  if  F,',  =  F^  =  0,  tlie  surface 
has  a  tangent  }dane  parallel  to  z  =  0,  and  if  the  condition  ,-  =  F  =  0  is 
also  satisfied,  the  surface  is  tangent  to  the  ./-//-plane.  Now  if  ■:  =  F(.r.  //) 
has  a  maxinuim  or  minimum  at  its  point  of  tangency  with  x  =  0,  tlie 
surface  lies  entirely  on  one  side  of  the  plane  and  the  point  of  tangency 
is  an  isolated  })oint  of  F(.i-,  ij)  =  0  ;  whereas  if  the  surface  has  a  mini- 
max it  cuts  througli  the  plane  z  =  0  and  the  point  of  tangency  is  not 
an  isolated  ])oint  of  Fix,  if)  =  0.  The  shape  of  the  curve  F=  0  in  the 
neighborhood  of  a  singular  i)oint  is  discussed  by  developing  F(,/-,  //) 
about  that  point  by  Taylor's  Formula. 

For  example,  consider  the  curve  F(x.  y)  =  xJ^  +  y^  —  J^'y'  —  l  W~  +  ?/'")  =  0  and 
the  surface  z  =  F(x,  y).    The  ci:)inmon  real  solutinns  of 

F;  =  iix^-2xy^-x  =  0.         f;^  =  S  y- -  2x^y  -  y  =  0,         F{x,y)  =  0 

are  the  singular  points.  The  real  solutions  of  F.^.  =  0.  F,^' =  0  are  (0.  0).  (1.  1), 
(1,  I)  and  of  these  the  first  two  satisfy  F{x.  y)  —  0  but  the  last  d(»es  not.  The 
singular  points  of  the  curve  are  therefore  (0.  0)  and  (1.  1).  The  test  (34)  of  g  55 
.shows  that  (0.  0)  is  a  maximum  for  z  =  F{x.  y)  and  hence  an  isolated  point  of 
F(x.  y)  =  0.  The  test  also  shows  that  (1.  1)  is  a  minimax.  To  discuss  the  curve 
F{x.  y)  =  0  near  (1,  1)  apply  Taylor's  Formula. 

0  =  F(x,  y)  =  I  (3  h-  -  8  hk  +  3  k-)  +  |  (0  h"^  -  12  h-k  -  12  hk-  +  C  }c^)  +  remainder 
:=  I  (3  COS-  0—8  sin  cp  cos  0  -f  3  sin-  0) 

-j-  /•  (cos-'  0  —  2  ci_)s-  0  sin  0  —  2  cos  0  sin-  0  -f  sin^  0)  -f  •  •  • . 


120  DIFFERE]^TIAL  CALCULUS 

if  polar  coordinates  h  =  rcos<^,  k  =  r  sin  0  be  introduced  at  (1,  1)  and  r^  be  can- 
celed. Now  for  very  small  values  of  r,  the  eciuation  can  be  satisfied  only  when 
the  first  parenthesis  is  very  small.    Hence  the  solutions  of 

3  -  4sin  2  0  =  0,         sin  2  0  =  |,       or       0  =:  24'"  \1V,  (i-P  42^', 

and  0  +  77,  are  the  directions  of  the  tan<;-ents  to  -F(.r,  //)=  0.    The  equation  F  =  0  is 

0  =  (U  —  2  sin  2  0)  +  r{c()>^4>  +  sin0)(1  —  1^  sin  2  0) 

if  only  the  first  two  terms  are  kept,  and  this  will  serve  to  sketch  F{x,  y)  =  0  for 
very  sm-.U  value,',  of  r,  that  is,  for  0  very  near  to  the  tangent  directions. 

58.  It  is  hnj)ortiint  to  obtain  conditions  for  tlie  niaxiinuin  or  mininnim 
of  a  function  z  =f(^.i',  y)  where  tlie  variables  ./■,  ;j  are  connec'ted  by  a 
relation  -/''(■'',  .'/)  =  0  so  tliat  z  really  becomes  a  function  of  x  aloiu;  or  // 
alone.  For  it  is  not  always  possible,  and  fre(|uently  it  is  inconvenient, 
to  solve  F(:r,  ?/)  =  0  for  either  variable  and  thus  eliminate  that  variable 
from  z  ^ /(,!-,  y)  by  substitution.  AVhen  the  \'ariables  x,  y  in  -:  =f(x,  y) 
are  thus  connected,  the  minimum  or  maximum  is  called  a  (■(insfrdbird 
III  Inlin II III  or  iiKixim inn  :  when  there  is  no  e(juati(m  F(.i-,  y)  =  0  between 
them  the  juininium  or  maximum  is  called  free  if  any  designation  is 
needed.*  The  conditions  art;  ol)tained  l)y  dilt'erentiating  -.'  =f(x,  y) 
and  F{x,if)=  0  totally  Avitli  respet't  to  ./'.    Thus 

^  _  £/'       9^'  ['H  _  ^ 
dx       Cx       Cij  dx 

and  Iffl-- 2^-^=0,         :^,5  0,    '     V=%  (5) 

CX  cy       cy  cx  dx'  ^  ^ 

Avliere  the  first  e(|nation  ai'ises  from  tlie  two  above  l)y  eliminating  (fy/dx 
and  tlu!  second  is  added  to  insnre  a  nnnimum  or  maximum,  are  tlie  con- 
ditions desired.  Note  that  all  singular  points  of  F(x,  y)  =  0  satisfy  tlie 
fii'st  condition  ithMitically,  but  that  tlie  jirocess  by  means  of  which  it 
Avas  obtained  excludes  such  ])()ints,  and  that  the  rule  cannot  be  exju'cted 
to  a})pl\'  to  tlieni. 

Another  method  of  ti'oating  the  problem  of  constrained  maxima  and 
minima  is  to  inti-oduce  c  iinf/fl/dirr  and  form  the  function 

r:  =  <i>{.r.  y)  =f(x,  y)  -j-  \F{x,  y),  X  a  niulti[ilier.  (»>) 

Now  if  this  function  ::  is  to  hav(^  a  free;  maximum  or  minimum,  then 

K  =/',  +  ^f;  =  0,       <i>;,  =,/;;  +  af;  =  o.  (7) 

These  two  e([uatioiis  takini  with  7'' =  0  constitute  a  set  of  three  fi'oui 
wliich  tlie  tlii'ce  values  ;/■,  //,  A  niay  lie  obtained  by  solution.    Note  that 

•The  nil  ji'cti\i'  ■■  relative '■  is  sdinetiiiies  iiseil  fur  eiiiistraineil.  ami  "  absnliite  "'  I'nr 
free:  hnt  file  term  "alisdlute"  is  best  kept  foi-  the  greatest  of  tiie  inaxiiiia  m-  least  nf 
tile  iiiiiiiina,  and  tlie  term  "  relatixe  "  lor  the  other  maxima  and  minima. 


r/0 

cF 

cF 

'^!/ 

: 

=  — 

+ 

=. 

0, 

dx 

cx 

'(^'J 

dx 

d-z 

s=o, 

F 

:^ 

0, 

dx^ 

PAKTIAL  DIFFEKENTIATIOX;   IMPLICIT  121 

A.  cannot  be  obtained  from  (7)  if  both  F',.  and  F'^^  vanish;  and  hence  this 
nietliod  also  rejects  the  sing-uhir  points.  That  this  method  really  deter- 
mines the  constrained  maxima  and  minima  of  /'(./■,  //)  subject  to  tlu; 
constraint  F{x,  //)  =  0  is  seen  from  the  fact  that  if  A.  be  eliminated  from 
(7)  tlie  condition  _/',' F,^  —f'^F',.  =  0  of  (o)  is  obtained.  The  new  method 
is  therefore  identic^al  with  the  former,  and  its  introduction  is  more  a 
matter  of  convenience  than  necessity.  It  is  possible  to  show  directly 
tliat  the  new  metliod  gives  thc^  constrained  maxima  and  minima.  Por 
the  conditions  (7)  are  those  of  a  free  exti'cme  for  the  function  <!>(,/•,//) 
which  de])ends  on  two  inde])en(lent  varial)les  (./■,  //).  >«'o\v  if  the  e(|ua- 
tions  (7)  l)e  solved  for  (,r,  //),  it  ap})ears  that  the  position  of  the  maxinuuu 
or  minimum  will  be  expressed  in  terms  of  X  as  a  ]);irameter  and  that 
conse(piently  tlu'  ])oint  (.''(A.),  //(A))  cannot  in  general  lie  on  the  curvi; 
^'^ {■''■)  //)  =  Oj  ^'^d.  if  X  be  so  detei'mined  that  the  point  shall  lie  on  tliis 
curve,  the  funtdjon  $(.'•,  //)  has  a  free  extreme  at  a  point  for  which 
F  ~  0  and  hence  in  particular  must  have  a  constrained  extreme  foi'  the 
]iarticular  values  for  Avhich  F(x,  if)  =  0.  In  speaking  of  (7)  as  the  con- 
ditions for  an  extreme,  the  conditicMis  wliich  shoidd  be  imposed  on 
tlie  secoiul  di'rivative  have  been  disregarded. 

Yor  example,  suppose  the  iiiaxiuiuin  I'adius  vt'ctor  from  the  oi'ighi  to  the  folium 
of  Descartes  Avere  desired.  Tlu;  problem  is  to  reuder/(,f,  y)  =  x-  -f-  y-  maxiuuim 
subject  to  the  condition  F{x^  y)  =  x''  -\-  if'  —  3  (txij  =  0.    Hence 

2 X  -1-  o  X  (/'  —  iry)  =  0,         2  //  -H  ;}  X  (//-  —  ((,/■)  =  0,         x"'  -f-  //■'  —  8  <txij  =  0 

or  2 .r  •  3  (//-  —  ax)  —  2  ;/  ■  'P,  (x-  —  iiy)  —  0,         x''  +  y"'  —  3  axy  =  0 

are  the  conditions  in  thi'  two  cases.  These  (_'(iuations  may  be  solved  for  (0,  0), 
(1  ^  (I,  ^l  ((),  and  sonu'  ima.ninary  values.  The  value  (0,  0)  is  singular  and  X  cannot 
be  determined,  but  the  })oint  is  e\idently  a  mininnuu  of  i'-  -f  ?/-  by  inspection.  The 
point  (H  ",  1^  <()  ,ij,ives  'K  =  —  11  a.  'I'hat  the  point  is  a  (relative  consti'ained)  maxi- 
nuun  of  X-  +  v/-  is  also  seen  by  inspection.  'J'here  is  Jio  need  to  examine  d-f.  In 
most  practical  problems  the  exaniiualion  of  the  conditions  of  the  second  order 
may  be  waived.  'I'liis  example  is  one  which  may  be  treated  in  jiolar  c(>ru-dinales 
by  the  oi'dinary  methods;  but  it  is  iKiteworthy  that  if  it  could  not  l)e  treated  that 
Vvay.  Ilie  mt'thod  of  solution  by  eliminatiuL;-  one  of  the  variables  by  solvinj;-  the 
cubic  7''(.'',  y)  =  0  would  be  una\;iilablc  and  the  methods  of  constrained  luaxima 
woulil  be  re(piired. 

EXERCISES 

1.  V>\  total  differentiation  and  division  obtain  dy/dx  in  these  cases.  Do  not 
substitute  in  (1).  but  use  the  method  by  which  it  was  derived. 

{a)  ((X-  +  2  !ixy  +  cy-  —  1  =;  0,  (/i)  /'  4-  y*  =  4  d-xy,  (7)  (cos/)"  —  (sin  ?/)•'•  -  0, 
(0)   (■'■-  +  .'/-')-  =  "-(■'■■-—//-),  (e)  <:■■■+  c"  =  2  xy,  (f)  ,r- -//"-  =  tau-i  .(■//. 

2.  obtain  the  second  derivative  d-y/dx-  in  Kx.  1  (a).  (^).  (e),  (j")  3>y  differcn- 
tiatiu';-  t'.ie  value  of  dy/dx  ol)taincd  above,    t'onniare  with  use  of  (:!). 


122  LUFFEKEXTIAL  CALCULUS 

3.  Prove  ^  =  -  ^'^'"^^-^  ~  ^  ^'■'■^'1/K,  +  ^'x'-^yy 


y 

4.  Yhu\  the  radius  of  curvature  of  these  curves  : 

(tr)  j-I  +  y'-  =  rf  J,  7?  =  3  (a.r.y)3,         (/?)  j^  +  i/^  =ai,  R  =  2  V{x  +  yf/a, 
(7)  b-x-  +  a-y-  =  «-6-,         (5)  xy-  -  u-{<t  —  x),         (e)   (ax)-  +  {hy)'i  =  1. 

5.  Find  y',  y'\  y'"  in  case  x^  +  2/^  —  3  nxy  =  0. 

6.  ICxtcnd  equations  (3)  to  (il)taiii  //'"  and  reduce  by  Ex.  3. 

7.  Find  tangents  parallel  to  the  j--axis  for  (/-  +  y'-)-  =  2  «- (/-  —  y-). 

8.  Find  tangents  i)arallel  to  the  ^-axis  for  (,r-  +  ij-  -\-  ax)'-  =  ((- {x-  +  y-). 

9.  If  //-  <ac  in  ax-  +  2lixy  +  r//-  +/,/•  +  f///  +  //  =  0.   circumscribe    about   the 
curve  a  rectangle  parallel  to  the  axes.    Check  algebraically. 

10.  Sketch  x"  +  y"  =  x-y-  +  \  (.r-  +  y-)  near  the  singular  point  (1,  1). 

11.  Find  the  singular  points  and  discuss  the  curves  near  them  : 

(a)  r^  +  //■•  =  3  Hxy,  (/3)   {x-  +  y-)-  =  2  «-  (.c-  -  //-), 

(7)  .f^  +  y '  =  2  (.f  -  ?/)^  (5 )  y'>  +  2  .f ;/-  =  ^•-  +  y*. 

12.  Make  these  functions  maxima  or  nnnima  subject  to  the  given  conditions. 
Discuss  the  work  l)oth  with  and  without  a  nuiltiplier: 

,    ,        a  h  ,  .         sinx      u 

(a) 1 ,     a  tan  x  -\-  h  tan  y  —  c.  Aiis.  =  - . 

ucosx       i;cos(/  siny       u 

(j3)  ./•-  +  y-,     ax-  +  2hxy  +  ry-  =/.  Find  axes  of  conic. 

(7)    Find  the  shoi'test  distance  from  a  point  to  a  line  (in  a  plane). 

13.  Write  the  second  and  third  total  differentials  of  F{x.  y)  —  0  and  compare 
with  (;!)  and  Kx.  5.    Try  this  method  of  cahaUating  in  l-"x.  2. 

14.  Show  that  F'llx  +  F^^ily  =  0  does  and  should  i;ive  the  tangent  line  to 
F{x.  y)  =  U  at  the  points  {.c,  y)  if  dx  =  |  —  /  and  dy  =  rj  —  y.  where  ^.  77  are  the 
coordinates  of  points  other  than  {x,  y)  on  the  tangent  line.  Why  is  the  e(iuation 
inapplicable  at  singular  points  of  the  curve  '.' 

59.  More  general  cases  of  implicit  functions.  The  ])roblem  of 
ini])licit  fiuu'tioiis  may  be  generalized  in  two  ways.  Li  ilic  iirst  place 
a  gretiter  number  of  \'ariables  may  oeciir  in  the  ftnietion,  ;is 

^'{■•■,  .'A-)  =  f>,  F{:'',  :i,  .-,•■•,  "j  =  0; 

and  the  ({iiestion  may  be  to  solve  the  equation  for  one  of  the  variables 
in  ter]us  of  the  others  and  to  determim'  the  jKirtial  derivatives  of  tlie 
chosen  de])endent  variable.  In  the  second  ])lai'e  there  may  be  several 
eqntitions  connecting;-  the  variables  ;ind  it  may  be  recjiiired  to  solve  the 
(Mutations  for  some  of  the  variables  in  terms  of  the  others  atid  to 
determine   the  partial  derivtitives  of  the   chosen  dependent  variables 


r,t' 

-  l'^'^\  - 

r'r 

cz 

nh 

—  = 

—  ■ — ^ 

and 

—  = 

— ^ 

C.r 

'  V'-''), 

/;' 

f// 

\'^//, 

PARTIAL  DIFFEPvEXTlATlOX;   PMPPICIT  123 

with  respect  to  the  independent  variables.  In  both  cases  the  formal 
differentiation  and  attempted  formal  solution  of  the  equations  for  the 
derivatives  will  indicate  the  results  and  the  theorem  under  which  the 
solution  is  proper. 

Consider  the  case  F(.v,  //.  z)  =  0  and  form  the  differential. 

dF(x,  y,  z)  =  F:j.r  +  F','1>J+  Fl'lz  =  0.  (8) 

If  ,-;  is  to  be  the  dependent  variable,  the  partial  derivative  of  z  by  x  is 
found  by  setting  di/  =  0  so  that  //  is  constant.    Thus 

-p  (9) 

z 

are  obtained  l)y  ordinary  division  after  setting  <li/  =  0  and  dx  =  0  re- 
spectively. If  this  division  is  to  be  legitimate,  F.,  must  not  vanish  at 
the  point  considered.  The  immediate  suggestion  is  the  theorem  :  If, 
when  real  values  (./•,^,  ij}}  are  chosen  and  a  real  value  z^^  is  obtained 
from  F(z,  x^,  i/^^  =  0  by  solution,  the  function  F(x,  >/.  z)  regarded  as 
a  function  of  three  inde})endent  variables  (./■.  //.  z)  is  continuous  at 
and  near  (x^,  //^^,  zj  and  has  continuous  tirst  ])artial  derivatives  and 
F'.(x^^,  l/^,  z^)  ^  0,  then  F(x.  //.  z}  =  ()  may  be  solved  uni(]uely  for 
z  =  cf>(x,  I/)  and  <^(.'',  //)  will  be  continuous  and  liave  partial  derivatives 
(9)  for  values  of  (./■.  //)  sufficiently  near  to  C'^^,  i/^'). 

The  theorem  isai^ain  proved  by  thf  Law  (if  tlie  ^Nleaii,  and  in  a  similar  manner. 

F(x,  y,  z)  -  F{.r,^,  y„.  zj  =  F(j,  y.  z)  =  (hV,.  +  kV',,  +  '-P"^).,„r  e/-,.vo  + e<-.  .„  + e^- 

As  Fl.  F[^,  Fl  are  continuous  and  F'.,(.r^^,  ;/„,  s,,)  j^  0.  it  is  pussilile  to  take  5  so 
small  that,  when  |/(  [<  5.  1^-|  <  5.  |/1<  5,  tlie  derivative  | /•';!>  rii  and  !'i':,'|<;u.  \F',^\<fji,. 
Now  it  is  desired  so  to  restrict  h,  k  that  ±  57-'.'  shall  iletermine  the  sign  of  the 
parenthesis.    Let 

I J  —  J„  I  <  i  m5/tx,         I  i'  —  2/o  I  <  i  w5/m,     then     |  //  F,'  +  A'Fj  |  <  mb 

and  tlie  signs  of  the  parenthesis  f(ir  (.c.  y.  z,,  +  5)  and  (,r,  y.  z,,  —  5)  will  be  opposite 
since  |F^'|>7?i.  Hence  if  (.r.  y)  be  held  fixed,  there  is  one  and  only  one  value  of  z 
for  which  the  parenthesis  vanishes  lietween  z,,  +  5  and  z„  —  5.  'I'hus  z  is  defined  as  a 
single  valued  function  of  [x.  y)  for  sufficiently  small  values  of  /(  =  x  —  x^J.  k  =  y  —  y,,. 

Also  -  =-  K(-ro  +  ^''-vo  +  ^'^--^.  +  ^i)      I  __  K^::^ 

h"    F:{x,,  +  eh.y,,  +  ek.z„  +  eii      k~    f:{.--) 

when  k  and  h  respectively  are  assigned  the  values  0.  The  limits  exist  when  A  =  0  or 
k  =  0.  Bm  in  the  first  case  I  =  Az  =  A,.2  is  the  increment  of  z  when  x  alone  varies, 
and  in  the  second  case  I  =  Az  =AyZ.  The  limits  are  therefore  tlie  desired  partial 
derivatives  of  z  by  x  and  y.  The  proof  for  any  number  of  variabhs  would  be 
similar. 


124  diffeep:xtial  calculus 

If  none  of  the  derivatives  F'^.,  F„',  F',  vanish,  the  equation  F{x,  y,  z)  =  0 
may  be  solved  for  any  one  of  the  variable*^,  and  formulas  lilve  (9)  will 
express  the  partial  derivatives.    It  then  appears  that 


V/-' 

\/ 

',l.r' 

\ 

cz 

■  €., 

r' 

K 

f: 

=  1, 

Z' 

)( 

^^-, 

). 

~  C.r 

C-, 

S 

1^~ 

K  ~ 

<Jz^ 

\  / 

V/,/A 

J 

;///\ 

CZ 

ex 

^.'/ 

dXj 

K 

!^!// 

K 

M 

— 

ex 

<^'J 

cz 

—  I 

(10) 


in  like  manner.  The  first  ('(juation  is  in  this  case  identical  Avith  (  1) 
of  §  2  V)ecause  if  //  is  constant  the  relation  F{.i\  //,  r:)  =  0  reduces  tn 
G(x,  z)  =  0.  Tlie  second  equation  is  new.  Ly  virtue  of  (10)  and  simi- 
lar relations,  the  derivatives  in  (11)  may  be  inverted  and  transformed 
to  the  riylit  side  of  the  equation.  As  it  is  assumed  in  tliermodynamics 
that  the  pressure,  volume,  and  temperature  of  a  given  simple  sul)stance 
are  connected  by  an  equation  F(p,  r,  T)  =  0,  called  the  characteristic 
etpiation  of  the  sid)stance,  a  relation  between  different  thermodynamic 
magnitudes  is  furnished  by  (11). 

60.   In  the  next  place  suppose  there  are  two  equations 

/•'(■'•,  .'A  ",.  '0  =  ^^  ^'C'-,!/,",  <■)  =  ()  (12) 

between  four  vai'iablcs.    T^'t  each  equation  be  differentiated. 
,/F  =  0  =  F//x  +  /•',////  +  f;,>/u  +  F/h-, 

(If;  =  0  =  (i//x  +  r/v/y  +  a^u  +  (;%/,-.  (1.3) 

If  it  l)e  desired  to  consider  //,  r  as  the  dependent  variables  and  ./■,  //  as 
inde])endent,  it  would  be  natural  to  solve  these  equations  for  the  dilfer- 
entials  (In  and  (/<•  in  terms  of  i/.r  and  fh/;   for  exam})le. 

(Fv;;  -  f;/;;.)  dx  +  i  f;/;;.  -  F;jr, ,  ./y  ., 

an  =  — —' ,     , (1.-) 

The  differential  i/r  would  have  a  different  numeratoi'  Imt  tlie  same  de- 
nominator. Tlie  solution  reijuires  /\6','  —  F',.fr'^  —  0.  This  suggests  llie 
desired  theorem  :  If  (i/^.^.  r^  )  ai'e  soluti(jns  of  F  =  0.  (i  =  0  c()n-es[ion(liiig 
to  (r^,  ?/j  and  if  F^fr^.  —  F^/F,  does  not  vanisli  for  the  values  (./■..  y  .  i/.  r  ), 
the  equations  F  —  0,  G  —  0  may  be  solved  for  >'  =  cfi(x,  //).  r  =  ibi.r.  if) 
and  the  solution  is  miique  and  valid  for  (./•.  //)  sufficiently  near  (.'■^.  y  ) 
—  it  lieing  assumed  that  /^"and  '■/  I'egarded  as  functions  in  four  variables 
are  continuous  and  have  continuous  first  ])artial  derivatives  at  and  iieai- 
(.'•-,.  //. ,  11^^,  rj  ;  moreover,  the  total  differentials  dc,  dr  are  given  by  ( 1.'3') 
and  a  similar  t'<piati(jn. 


PARTIAL  DIFFEKEXTIATTOX;   I.ALFLICIT  125 

The  proof  of  this  theorem  may  be  deferred  (§  G4).  Some  observations 
should  be  made.  Tlie  e(piations  (13)  may  he  solved  for  any  two  \-ari- 
ables  in  terms  of  the  other  two.    The  partial  derivatives 

d>i(.r,//)  r>'(.r,r)  C.r(u.r)  C.r  ()/,//) 

;; J  7 }  ;. '  ■ 7- ( -L  -i ) 

CJ'  C.I-  Cll  cu  ^       ' 

of  V  ])y  .r  or  of  x  l)y  //  Avill  naturally  dei)end  on  Avhether  the  solution 
for  u  is  in  terms  of  (./•,  //)  or  of  (./ ,  r),  and  the  solution  for  ./•  is  in  i^ii.  r) 
or  (;/,  if).  ]Moreovei',  it  must  not  be  assununl  that  du /dx  and  cx Jcii  are 
reciprocals  no  matter  Avhich  meaning  is  attached  to  each.  In  obtaining 
relations  between  the  derivatives  analogous  to  (10),  (11),  the  values  of 
the  derivatives  in  terms  of  the  derivatives  of  F  and  G  niiiy  be  found  or 
the  equations  (12)  may  first  be  considered  as  solved. 

Thus  if  u  =  (p  {x,  y),  du  =  (p/lx  +  4>^/ly, 

V  =  i^  (.(•,  y),  dv  =  x^/lx  +  xi^^/hj. 

Then  dx  =  ^-"— ^^ ,  dy  =  -^^^ f^ 

,                                  fx                "AJ                       cx              -  4>', 
and  —  = ,  -     = ,  etc. 

Cll    0>;  -  ,p\p;.        cv    0;^,; _  0^^; 

cu  cx      cv  cx 

Hence  +       -  -  =  1 ,  (15) 

cx  cu      cx  cv 

as  may  be  seen  by  direct  substitution.  Here  u,  v  are  expressed  in  terms  of  x.  y  for 
the  derivatives  m^',  r,' ;  and  x,  y  are  considered  as  expressed  in  terms  of  u,  v  for  the 
(U'rivatives  j,^,  x^.. 

61.  The  questi(jns  of  free  or  constrained  maxima  and  minima,  at  an}' 
rate  in  so  far  as  the  determination  of  the  conditions  of  the  first  order  is 
concerned,  may  now  ])e  treated.  If  /■'(./•,  //,  r.)  =  0  is  given  and  the  max- 
ima and  minima  of  r:  tis  a  function  of  (./•,  //)  are  wanted, 

F;  (.'■,  ;/,  z)  =  0,  f;  (x,  !i,  rS)  =  0,  F(x,  y,  ^)  =  0  (1 0) 

are  three  equations  which  may  V)e  solved  for,/-,  //, .-.'.  If  for  any  of  these 
solutions  the  derivative  F_'  does  not  vanisli,  the  surface  .v  —  4>(.r.  //)  has 
at  that  point  a  tangent  plane  parallel  to  ,-;  =  0  and  there  is  a  maximum, 
minimum,  or  minimax.  To  distinguish  between  the  possibilities  further 
investigation  must  be  made  if  necessary  ;  the  details  of  such  an  investi- 
gation Avill  not  be  outlined  for  the  reason  that  special  methods  are 
usually  availal)le.  The  conditions  for  an  extreme  of  1/  as  a  function  of 
(,/■,  )/)  defined  implicitly  by  the  ecpiations  (13')  are  seen  to  be 

Fr;;.  _  /-^y;;  =  o,    f;g;.  -  f;/;;,  =  o,    f=  o,    g  =  o.        (17) 

The  four  ecpiations  may  be  solved  for  ,/•,  y,  k,  c  or  merely  for  ,/■,  //. 


120  DIFFEKEXTIAL   ("AIJ'ULUS 

Suj)})()se  that  tlie  maxima,  minima,  and  lainimax  of  n  ^=f(x,  //,  z)  su]> 
jeet  eitlitr  to  one  e(|uation  F{.r,  //,  z)  =  0  or  two  equations  F{.r.  //,  ,-.)  =  0, 
6'(./',  //,  ,v)  =  0  of  constraint  ai'e  desired.  Xote  tliat  if  only  one  equation 
of  constraint  is  imposed,  the  function  u  =/(./■.  tj,  z)  l)ecomes  a  function 
of  two  varial)les  ;  whereas  if  two  equations  are  imposed,  tlie  function  v 
really  contains  only  one  varial)le  and  the  question  of  a  minimax  does 
not  arise.    The  mrtliod  nf  intilt'qdicrs  is  again  employed.    Consider 

^(■'■,  y, -)=/+A/-^     or     ^=fJrXF+ixG  (18) 

as  the  case  may  be.    The  conditions  for  a  free  extreme  of  $  are 

$;  =  0,         *;  =  0,         cj>:  =  0.  (19) 

Tliese  three  equations  may  l)e  solved  for  the  coordinates  ./•,  //,  z  which 
will  then  he  expressed  as  functicjns  of  X  or  of  A  and  ^i  according  to  the 
case.  If  then  A  or  A  and  /x  be  determined  so  that  (./■,  //.  z)  satisfy  F  =  0 
or  i*"  =  0  and  G  =  0,  the  constrained  extremes  of  v  =^f{.r.  //,  z)  will  Ije 
found  exce})t  for  the  examination  of  the  conditions  of  higher  order. 

,\s  a  ])riil)le!ii  in  coiistriuned  maxima  ami  minima  let  the  axes  (if  the  section  of 
an  ellipsdid  by  a  plane  through  the  origin  l)e  ik'termined.    Form  the  function 

*  =  X-  +  //-  +  2-  +  X  ("^  +  -f,  +  -.  -  n  +  M  {Ix  +  my  +  nz) 

by  adding  to  x-  +  ;/-  +  2-,  which  is  t<  i  be  made  extreme,  the  equations  of  the  ellipsoid 
and  plane,  which  are  the  equations  of  constraint.    Then  apply  (IH).    Hence 

(/-       2  \r       -1  (f-       2 

taken  witli  tlie  equatiims  of  ellipsnid  and  plane  will  determine  x.  y.  z.  X.  /i.  If  the 
equations  are  nuiltiplied  by  /.  //.  z  and  reduced  by  the  equations  of  plane  and 
ellipsoid,  the  solution  for  X  is  X  =—  r-  ~—  (x-  +  y-  +  z-).  The  three  equations 
then  become 

1  u.J(l-  1     ainh-  1     unc-  .  ,       , 

X  = ,         y  = .         z  = ,         with    It  +  my  +  nz  —  0. 

2  ?■-  —  o'-  2  r-  —  h-  2  /•-  —  ';- 

\-(i-  m'-h-  n'-c- 

Hence ;H h —  =0     determines  r-.  (20) 

/•-  —  (i-       r-  —  b-       ;■-  —  c- 

Tlie  two  rniits  for  /•  are  the  major  ami  minor  axes  of  the  ellipse  in  which  the  jilane 
cuts  the  ellipsoid.    The  suVistinuion  of  /.  y.  z  above  in  the  ellips(jid  determines 


-        /      "/      \-       /     h)i,      \-       I     rii      \-        .  x-        if-       z-        ,  ,-, 


Now  when  (20)  i>  snlvi'd  for  any  particular  root  /•  and  the  value  of  fx  is  found  by 
(21  ).  1  !i-  actual  coordinates  s.  y.  z  of  the  extremities  of  the  axes  mav  lie  found. 


PARTIAL  1)1FFE1{E^'TIAT10X;   IMPLICIT  127 

EXERCISES 

1.  Obtain  the  partial  derivativL-.s  cif  z  l)y  x  and  g  directly  from  (8)  and  not  by 
substitution  in  (9).    Wliere  does  the  solution  fail  ? 

(a)  ^  +  J^  +  ?:  =  1,  (^)j  +  y+.=     l_, 

a-      b-      c-  xijz 

(7)  (•?-  +  .'/-  +  z-)'  =  n-x-  +  l)-y-  +  C-Z-,         (5)  xyz  -  c. 

2.  Find  the  second  derivatives  in  Ex.  1  [a),  (/3),  (5)  by  repeated  differentiation. 

3.  State  and  prove  the  theorem  on  the  solution  of  F{x.  ij,  z,  u)  =  0. 

4.  Show  that  the  product  ctpEr  of  the  coefficient  of  expansion  by  the  modulus 
of  elasticity  (§  52)  is  equal  to  the  rate  of  rise  of  pressure  with  the  temperature  if 
the  volume  is  constant. 

5.  Establish  the  proportion  E,^  -.  Et  =  C), :  C,-  (see  §  52). 

6.  If  I  {x,  y,  z,  u)  =  0,  show ^ =  1, =1. 

ex  cy  cz  cii  cx  cii 

7.  Write  the  ecjuations  of  tangent  plane  and  normal  line  to  F{x.  y.  z)  =  0  and 
find  the  tangent  Y)lanes  and  normal  lines  to  Ex.  1  (/i),  (5)  at  x  =  1.  y  =  1. 

8.  Find,  by  using  (13),  the  indicated  derivatives  on  the  assumption  that  either 
ar,  y  or  h,  v  are  dependent  and  the  other  pair  independent  : 

{ix)  ir'  +  r"'  +  x''  —  3y  =  0,         ir  +  v"'  +  y"  +  ?>x  =  0,         k'..  u'^.  m,',^,  r,',' 
(P)  x  +  y  +  u  +  V  =  (I,  X-  +  y-  +  u-  +  v-  =  h,  x^.  mJ,  t,/,  v'.y 

(7)  Find  dy  in  botli  cases  if  x.  v  are  independent  variables. 

9.  Prove  ^  ^  +  ^  ^^  =  0  if  F{x,  y,  u,  v)  =  0,  G  (x,  y,  u,  v)  =  0. 

cx  ell         cx  CK 

10.  Find  du  and  the  derivatives  u[..  ",^.  u'.  in  case 

J'  +  y''  +  2-  ~  uv,         xy  =  u-  +  r-  +  vf-,         xyz  =  uvw. 

11.  If  E(,r.,  y.  z)  —  0,  (t{x,  y,  z)  =  0  define  a  curve,  show  that 

■g-Jp  _  y  -  iJn z-  g(i     

is  the  tangent  line  to  the  curve  at  (.r,,.  y„.  z,,).    Write  the  normal  plane. 

12.  P'ornuilate  the  problem  of  implicit  functions  occurring  in  Ex.  H. 

13.  Find  the  perpendicular  distance  from  a  point  to  a  plane. 

14.  The  sum  of  three  positive  nundaers  is  x  +  y  +  z  —  X.  where  J\"  is  given. 
Determine  x.  y.  z  so  that  the  product  xi'yiz''  shall  be  maxinuun  if  p.  7.  r  are  given. 

Ans.  X  :  y  :  z  :  X  =z  p  :  q  :  r  :  {p  +  q  +  r). 

15.  The  sum  of  three  positive  numbers  and  the  sum  of  their  squares  are  both 
given.    Make  the  i)roduct  a  maximum  f)r  mininuim. 

16.  The  surface  (x-  +  y-  +  z'-)-  =  ax-  +  hy-  +  cz-  is  cut  by  the  plane  lx  +  ]iu/  +  uz=.0. 

Find  the  maximum  or  mininmm  radius  of  the  section.  .Ins.     > =  0. 

^  r-  —  (I 


128  DIFFERENTIAL   CALCULUS 

17.  In  ca.se  F(x,  ?/,  w,  v)  =  0,   G'(x,  ?/,  w,  r)  =  0  consider  the  dilferentials 

7  ^"     7  ,      f'^      7  7  ^'^     7  ,      ^-^^     7  7  ^ ''/     7  ,      ^ !^     7 

au  =  —  «x  -I ay,        ax  =  —  an,  -\ dv,        ay  =  —  du  ^ dv. 

ex  cy  cii  cv  cu  cv 

Substitute  in  the  lirst  from  the  hist  two  and  obtain  relations  like  (1-j)  and  Ex.  9. 

18.  If  f{x.  y,  z)  is  to  be  niaxinuuu  or  uiininuun  subject  to  the  constraint 
F(x,  y,  z)  =  0,  show  that  the  conditions  are  that  dx  :  dy  :  dz  =  0  A) :  0  are  indeter- 
minate when  their  solution  is  attempted  from 

f)lc  +  f',(hj  +  f'jlz  =  0     and     F'dx  +  F'/ly  +  Fjlz  =  0. 

From  what  jjeometrical  considerations  should  this  I)e  olnious  '.'  Discuss  in  connec- 
tion with  the  problem  of  inscribing  the  maxinuun  rectangular  parallelepiped  in 
the  ellipsoid.    These  e(piations, 

dx  :  dy  :  dz  ^f^F^  -  f:j%  :fX-KK  ■CK-J'ijK  =  0^0:0, 
may  sometimes  Ije  used  to  advantage  for  such  prol)lenis. 

19.  (Jiven  the  curve  F{x,  ;/,  z)  =  0,  G' (x,  ?/,  z)  =  0.  Discuss  the  conditions  for 
the  highest  or  lowest  points,  or  more  generally  the  points  where  tlie  tangent  is 
parallel  to  z  =  0,  by  treating  u  =/(x.  y,  z)  =  z  as  a  maxinuun  oi'  minimum  sub- 
ject to  the  two  constraining  (Mjuations  F  —  0,  G  —  0.  Siinw  that  the  condition 
F^.G'  —  F'Cy.  which  is  thus  obtained  is  eciuivalent  to  setting  dz  —  0  in 

Fjlx  +  F',^dy  +  Fjlz  =  0     and     Gjlx  +  G'(ly  4-  Gjlz  =  0. 

20.  Find  the  highest  and  lowest  points  of  these  curves  : 

(a)  x"  4-  y~  =:  2-  +  1.   x  +  y  -\--2,z=.  0,         {p)  ''^  +  ■'-  +  '"-  =  1,  Ix  +  wy  +  nz  =  0. 

a-      h-       c- 

21.  Show  that  F%Jx  +  F^/ly  +  Fjlz  =  0,  witli  dx  :=  ^  -  .'■.  '///  =  •>?  —  //,  dz  =  f  —  z, 
is  the  tangent  plane  to  tlie  surface  F{x,  y,  z)  —  0  at  (./:.  //.  z).    Apph'  to  Ex.  L 

22.  (iiveu  F(x,  y.  u.  r)  -~  0,  G' (x,  //,  »,  v)  =  0.    obtain  the  (M|uations 

cF      (Fell      cFcv       ^  cF      cFcu       cFcv 

-  4-  -  +  —   -  =  0,  + +  =0, 

ex       eu.  cx       cv  ex  cy       cu  ey       co  cy 

cG       cGcu       cG  cv       ^  (G       cG  cu       cG  cV 

—  + +       —  =  0,       —  + +  —  .-  =  <!, 

cx       cu  ex       cv  ex  cy       eu  ey       ev  cy 

and  explain  their  signilicauce  as  a  sort  of  pai1ial-total  differentiation  of  /<' =  0 
and  G  =  0.  Find  u'^.  from  them  and  compare  witli  (K)').  Write  similar  e(]uatinns 
where  x,  y  are  considered  as  functions  of  (».  v).  Hence  pr(i\'e,  and  compai'c  with 
(15)  and  Ex.  0, 

cu  cy       eV ey   _              cu eX       cv cX  _ 
— \-     -       —  1,  7 +  -  —  0. 

ey  eu      c y  ev  ey  eu      c y  co 

23.  Show  that  the  differentiation  with  respect  tn  x  and  y  of  the  four  equations 
under  Ex.  2'2  leads  to  eiglit  ecjuations  fi-om  which  the  eight  derivatives 

c-u  e-u  e-u  c'-u  c'-v  e'-v 

cx-  cxey  eyex  cy-  f.c-  cy- 

mav  be  olilained.    Show  thus  that  fnrmallv  u''„  ^~  u''.. 


PARTIAL  DIFFERENTIATION;  IMPLICIT  129 

62.  Functional  determinants  or  Jacobians.   Let  t^vo  functions 

'<  =  ^(:';  I/),  '•  =  '/'(•'•,//)  (22) 

of  two  independent  variables  l)e  given.  Tlie  continuity  of  the  functions 
and  of  tlieir  first  derivatives  is  assumed  throughout  tliis  discussion 
and  "will  not  be  mentioned  again.  Suppose  that  there  were  a  relation 
F(u,  r)  =  0  or  /'\<^.  i/')  =  0  betAveen  the  functions.    Then 

F(<i>,^)  =  o,     f:<j>:+f;^:^  =  o,     F:<i>'„  +  F:.^;  =  o.    (23) 

The  last  two  equations  arise  on  differentiating  the  first  with  respect  to 
X  and  y.    The  elimination  of  F^^  and  F'^  from  these  gives 


•^i'/'.v  -  4>i,'Px  = 


^^^'''■^-./(:^)=0.  (24) 


^(■'•jy}       V,  u 


The  determinant  is  merely  another  way  of  writing  the  first  expression  ; 
the  next  form  is' the  customary  short  Avay  of  Avriting  the  determinant 
and  denotes  that  the  elements  of  the  determinant  are  tlie  first  deriva- 
tives of  }i  and  V  with  respect  to  .'■  and  //.  This  determinant  is  called  the 
functional  (h'tei'mlnant  or  Jdcohlini  of  the  functions  ii,  r  or  <^,  i^^  with 
respect  to  the  varial)les  x,  //  and  is  denoted  l)y  ./.  It  is  seen  that :  If 
there  is  a  functioncl  n  hithni  F((j),  i/^)  =  0  Jictu-cen  tiro  finictiotts,  tlif 
Jacohian  of  the  functions  rotiishes  identic// Ity^  that  is,  vanishes  for  all 
values  of  the  variables  (x,  //)  under  consideration. 

Conversely,  iftheJ('c/,//i//n  r//nishes  identicalli/  oi-cr  ti  ttr/j-zJim/nisi/m/il 
region  for  (x,  i/),  the  fnn/iions  ore  connected  hi/  <i  fun/ilon/il  rcl/iti/m. 
For,  the  functions  v,  v  may  be  assumed  not  to  reduce  to  mere  constants 
and  hence  there  may  lie  assumed  to  l)e  points  for  which  at  least  one  of 
the  partial  derivatives  ^,',  </),^,  i//'.,  i//,^  does  not  vanish.  Let  ^',.  be  the 
derivative  which  does  not  vanisli  at  some  particular  point  of  the  region. 
Then  u  =  <^(.>',  rj)  may  be  solved  as  x  =  xi^i-,  y)  in  the  vicinity  of  that 
point  and  the  result  may  be  substituted  in  v. 

<^->'  cu  ex  ^      ^>'        1        ,    ,         ,    ,  /<^,,. 

But  ^=-^-     and     —  =  —  {<f>4-^xl>:)  (24') 

Cy  cy  cu  cy       <^,  •     ^-^       ^      ■" 

by  (11)  and  sul_«titution.  Thus  cr fey  =  .//c^.,'. ;  and  if  ./  =  0,  then 
cv Icy  =  0.  This  relation  holds  at  least  throughout  the  region  for  which 
c^'.  4^  0,  and  for  points  in  this  region  cr  jcy  vanishes  identically.  Hence 
r  does  not  dex)end  on  //  but  becomes  a  function  of  ."  alone.  This  es- 
ta1)lishes  the  fact  that  e  and  a  are  functionallv  connected. 


130  DIFFERENTIAL   CALCULUS 

These  considerations  may  be  extended  to  otlier  cases.    Let 
n  =  <f>(.r,  I/,  z),  r  =  xp{x,  y,z),  "' =  X^''''  //: -)• 

If  there  is  a  functional  relation  /•"('/,  r,  ir)  =  0,  differentiate  it. 


(25) 


(26) 


K¥.,  +  ^''.4'u  +  ^'.rXi,  =  0,       <i>'.   ^'.    v,:  =  0, 

or  T ^=  T =  ./  =  u. 

c (./',  //,  -:)        c (,'•,  ij,  ■:) 

The  result  is  obtained  by  eliminating  Fj,  F,'.  F,^,  from  the  thi'ee  equations. 
The  assum])tion  is  made,  here  as  al>ove,  that  FJ,  F,!,  F,^.  do  not  all  vanisli ; 
for  if  they  did,  the  three  equations  would  iu)t  im})ly  ./ =  0.  On  the 
other  hand  their  vanishing  would  imply  that  F  did  not  contain  //.  r.  ir. 
—  as  it  must  if  there  is  really  a  relation  fyetwccn  them.  And  now  con- 
vei'sely  it  may  be  shown  that  if  ./  vanishes  identically,  tliere  is  a  func- 
tional I'clation  lu'twcen  //,  c.  "■.  Hence  again  f]ic  /im'ssm'//  /'//_'/  suijicii-nf 
ro)iiIifir)iis  flidf  fill-  tltri'i'  finirt'ujns  (25)  he  fanctlonulhj  conncctnil  is  that 
til eir  Jacdfii'i  11  I'd n  <sli . 

The  pniof  nf  tlic  (.■iniviTse  part  is  about  as  Ijcforc.  It  may  lie  assumeil  that  at 
least  one  of  the  derivatives  of  u.  v.  ir  or  (p.  ■^.  y(^  liy  .r.  ;/.  z  does  not  vanish.  Let 
0^'  :^  0  be  that  derivative.  Then  u  =  (p(.r.  ;/.  z)  may  be  solved  as  x  ~  w{u.  y.  z) 
and  the  result  may  l>e  substitute(l  in  r  and  id  as 

r  =  i//  {.r.  II.  z)  =  ^■p  (oj.  y.  z).         V-  =  X  (■'-•  .'/•  z)  =  X  i^-  V-  z)- 
Next  the  Jaeobian  of  r  and  c-  relative  to  //  and  z  mav  be  written  as 


, (X  ,  , (X  , 

"A,-  .-  +  •'/■,/     X.,-  V-  +  X.i 
cy        '  cy         ' 


f  r 

(>r 

f> 

('J 

(V 

cu- 

cz 

cz  i 

'/'.-.'     x,t 


<>. 


i., 


X: 


-0, 

/^.: 

x'l 

1 

i',, 

-<P;, 

h'A 

-  't'. 

/r.- 

X,: 

+  x;  ,, 

~-    '^c/  'Pj- 

■    X., 

'!'', 

4>:, 

'^'ll 

J 

+  f 

\  +  ^''' 

= 

\  X-' 

'}'■: 

'P', 

'•fz 

'p. 

As./  vanishes  identiralh',  the  Jacobjan  of  r  and  >'•  ex]iressed  as  funetions  of  ?/.  z, 
also  vanishes.  Ilenee  by  the  case  previously  iliseiissetl  there  is  a  functional  rela- 
tion F(v.  ir)  =  0  inilependent  of  //.  .-/  ;  and  as  v.  n:  ikjw  contain  ?(.tliis  relation  may 
be  considereil  as  a  fuiK'tional  relation  betwi/en  u.  v.  i'\ 

63.  If  in  (22)  the  variables  i/,  r  be  ;issigne(l  c()n>tunt  values,  tlie 
e(|uations  define  two  curves,  and  if  /'.  r  be  assigiuMl  a  series  of  sucli 
values,  file  equations  (22j  define  a  network  of  ctirx'cs  in  some  part  of  the 


PARTIAL  DIFFERENTIATION:   BIPLICIT 


131 


a'//-plane.  If  there  is  a  functional  relation  v  =  t^i''),  that  is,  if  tlie 
Jaeobian  vanishes  identically,  a  constant  value  of  c  implies  a  constant 
value  of  u  and  lience  the  locus  for  which  r  is  constant  is  also  a  locus 
for  which  v  is  constant ;  the  set  of  r-curves  coincides  with  the  set  of 
^-curves  and  no  true  network  is  formed.    This  >, 

case  is  uninteresting.  Let  it  be  assumed  that 
the  Jaeobian  does  not  vanish  identically  and 
even  that  it  does  not  vanish  foi'  any  point  (,/■,  //) 
of  a  certain  region  of  the  .///-iJane.  The  indi- 
cations of  §  GO  are  that  the  ecpiations  (22)  may 
then  be  solved  for  .r,  //  in  terms  of  u,  r  at  any 
})oint  of  the  region  and  that  there  is  a  pair  of 
the  curves  tlirough  each  point.  It  is  then  pro})er  to  consider  (u,  y)  as 
the  coordinates  of  the  points  in  the  region.  To  any  point  there  corre- 
spond not  only  the  rectangular  coordinates  (./■,  >/)  but  also  the  n/rri- 
I'lnedr  ruurdindtf's  (ii,  r). 

The  equations  connecting  the  rectangular  and  cairvilinear  coordinates 
may  be  taken  in  either  of  the  two  forms 

n  =  (/>(>•,  y),         r  =  ijy(.r,  ;j)      or      ./•  =J\a,  r),         y  =  g{u,  r),     (22') 

each  of  which  are  the  solutions  of  the  other.    The  Jacobians 


Y 

l\ 

0 

A' 

J(^  ../  ^1=1 


(27) 


are  reciprocal  each  to  each  ;  and  this  rela- 
tion may  be  regaitled  as  the  analogy  of 
the  relation  (4)  of  §  2  for  the  case  of 
the  function  ii  =  c^(.r)  and  the  solution 
w  =  f(ij)  =  <f>~^(i/)  in  the  case  of  a  single 
variable.    The  difff^nnfi'il  of  (/re  in 


(x  +  di-x,  y  +  cly'y) 

(u,  v+dv) 

(x  +  dx.p+dy) 

{u  +  du,  v+  dv) 

v+dv 


(x  +  dux,  i/+diiv) 
(u  +  du.  v) 


.Cd  \Clt 


d.^  =  d.r-  -f  df  =  Kdir  +  2  Fdiidr  +  Gdr"-, 

fr 


X 

(28) 


C.r  C.I-  C lie  11 

Cii  cr        Cii  cr  ' 


ca         \cc 


The  d  life  rent  i' 1 1  i,f  nrcn  included  between  two  neighboi'ing  ^/-curves  and 
two  neiuhborinsj'  r-curves  mav  Ije  written  in  the  form 


dA  =j{''^\dddr  =  dudr -r-J         '' 


These  statements  will  now  ])e  proved  in  detail. 


(29) 


132 


DIFFERENTIAL   CALCULUS 


To  prove  (27)  write  out  the  Jacobians  at  length  and  reduce  the  result. 


u,  V 

,x,  y. 


•c,  y 


ell 

cv 

cX 

cy 

cx, 

cx 

Cll 

cu 

Cll 

cv 

cx 

cy 

cy 

cij 

cv 

cv 

cu  cx 

CX  cll 


cu  CX        C  V  cX 
cy  cu      cy  cv 


cv  cx  cu cy 
cx  cv  cx  cll 
cu  cy 


cy  cu 


+ 


cvcy 

1 

0 

cx  cv 

cv  cy 
cy  cv 

0 

1 

where  the  rule  for  multiplying  detenuinants  has  been  applied  and  the  reduction 
has  been  made  by  (15),  Ex.  9  above,  ami  similar  fonnulas.  If  the  rule  for  multi- 
plying determinants  is  unfamiliar,  the  Jacobians  may  be  written  and  multiplied 
without  that  notation  and  the  reduction  may  be  made  by  the  same  f(jrnuilas  as 
before. 

To  establish  the  formula  for  the  differential  of  arc  it  is  only  necessary  to  write 
the  total  differentials  of  dx  and  dy,  to  square  and  add,  and  then  collect.  To  obtain 
the  differential  area  between  four  adjacent  curves  consider  the  triangle  determined 
by  (u,  v),  (u  +  du,  y),  (ii,  v  +  dv),  which  is  half  that  area,  and  double  the  result. 
The  determinantal  form  of  the  area  of  a  triauulc  is  the  best  to  use. 


dA=2- 


d„x    duy 
dyX     d^y 


dv 


'^Uu 

CX 

cy 

— 

cu 

CM 

cu 

cy  , 

CX 

cy 

—  cv 

— 

cv 

cv 

cv 

dudv. 


The  subscripts  on  the  differentials  indicate  which  variable  changes  ;  thus  J„x,  duy 
are  the  coordinates  of  (w  +  du,  v)  relative  to  (w,  v).  This  method  is  easily  extended 
to  determine  the  analogous  quantities  in  three  dimensions  or  more.  It  may  be 
noticed  that  the  triangle  does  not  look  as  if  it  were  half  the  area  (except  for  inlin- 
itesimals  of  higher  order)  in  the  ligure  ;  but  .see  Ex.  12  below. 

It  should  be  reniarkfd  tliat  as  the  differential  of  area  r/.l  is  usually 
considered  positive  Avhen  die  and  (7a  are  positive,  it  is  usually  ])etter  to 
rei)lace  ./  in  (29)  by  its  absolute  value.  Instead  of  rei^ardiii::^'  ('//,  r)  as 
curvilinear  coordinates  in  the  ,/;//-})lane,  it  is  ]»ossil)le  to  plot  theui  in 
their  own  ///--plane  and  thus  to  establish  l)y  (22')  a  tr<insf(n'in'if((i)i  of 
the  a-//-plane  over  onto  the  ///--plane.  A  small  art^a  in  the  .''//-plane  tlien 
becomes  a  small  area  in  the  ///--plane.  If  J  >  0,  the  transformation  is 
called  direct ;  but  if  ,/  <  0,  the  transformation  is  calli'd  perverted.  The 
significan(;e  of  the  distinction  can  be  made  clear  only  when  the  (pies- 
tion  of  tlie  sig-ns  of  areas  has  been  ti'eated.  The  transformation  is  called 
conforiiHiI  when  elements  of  arc;  in  the  neig-hl)orhood  of  a  ])()int  in  tlie 
.r/y-plane  are  jtroportional  to  tlie  ehunents  of  ar(^  in  the  neighl)orhood  of 
the  corresponding  point  in  the  ///--plane,  that  is,  when 


d/'  +  ////  =  A-  (////-  -f  ///--)  =  /.v/o--. 


(30) 


PAKTIAL  DIFFEKENTIATIOX ;   i:\IPLIOIT 


133 


For  in  this  case  any  little  triang-le  will  be  transformed  into  a  little  tri- 
angle similar  to  it,  and  hence  angles  will  be  nnclianged  by  the  transfor- 
mation. That  the  transformation  be  conformal  re(}ui]'es  that  F  —  0  and 
E  =  G.  It  is  not  necessary  that  E  =  G  =  /.•  be  constants ;  the  ratio  of 
similitude  may  be  different  for  different  points. 

64.  There  remains  outstandings  the  proof  tliat  equations  niay  be  solved 
in  the  neighl)orhood  of  a  point  at  whi(*h  the  Jacobian  does  not  vanish. 
The  fact  was  indicated  in  §  60  and  used  in  §  63. 

Thkohkm.    Let^y  e(}uations  in  n  -\-  p  variables  be  given,  say, 

^(•'V  •'■-  •  •  •'  ^■"+.)  =  0.        7^  =  0,  . . .,  7-;,  =  0.  (81) 

Let  the  />  functions  be  soluble  for  .r^^,  .r.^,  •  •  •,  .r^^^  when  a  particular  set 
■"*"(;' +i)o'  ■  ■  ■'  •'^(n  +  p)o  ^^  ^^^^  other  71  variables  are  given.  Let  the  functions 
and  their  first  derivatives  be  continuous  in  all  the  n  +2^  variables  in  the 
neighborhood  of  (,/■  ,  ,r, ,  •  •  •,  ■i'(„  +  „^ ).    Let  the  Jacobian  of  the  functions 

o  \    lo"      io"  ■      (II  +  I'm' 

with  respect  to  ./'^  ./•.,,  ■  •  • ,  x^^, 


r,, 


dJ-\ 

dF, 

cy 

d.r^ 

'^■\. 

..^ 

C,/' 

C.I- 

^0, 


(32) 


'  (»+p)o 


fail  to  vanish  for  the  particular  set  mentioned.  Then  the  ])  equations 
may  be  solved  for  the  /J  variables  .r^,  ./•„,  •  •  •,  .r^,,  and  the  solutions  will  lie 
continiious,  unicjuc,  and  diffcrentiable  witli  continuous  first  partial 
derivatives  i'or  all  values  of  ■'),mi,  ■■■,  ■''„  +  ,<  sufficiently  near  to  the 
values  ;'V,,+i)„.  •••,  '■''(n  +  /Oo- 

TiiKoKKM.  The  necessary  and  sufficient  condition  that  a  functif)nal 
relation  exist  between  jt  functions  of  //  vai'iables  is  tliat  tlie  -lacobian 
of  the  functions  with  respect  to  the  variables  shall  vanish  identically, 
that  is,  for  all  values  of  the  variables. 

The  proofs  of  tliesc  tlicorcins  will  naturally  lie  !:;iv(Mi  by  niathoniatiral  iinluclioii. 
Ivicli  of  the  theorems  has  been  proved  in  the  simplest  cases  and  it  remains  only  to 
show  that  the  theorems  are  true  for  p  functions  in  ease  tliey  are  for  ]>  —  ^ .  I^xpand 
the  determinant  ,f. 


cx, 


F. 

ex., 


f'F, 


^^'  +  '/.v^  +  ---  +  -/..l'^' 


J, 


,  J^,.  minoi'.- 


l"or  the  tirst  theorem  J  ^  0  and  hence  at  least  one  of  the  minors  J^,  ■  ■  •.  J^,  nnist 
fail  to  vanish.  Let  that  one  be  ,[^,  which  is  tlu'  Jacobian  of  7\,,  •  •  • .  Fj,  with  res})e(.'t 
tij  x.^,  •  •  •,  X/,.  By  the  assumption  that  the  theorem  liolds  for  the  case  p  —  1.  these 
p  —  1  ecjuations  may  be  solved  for  x.,.  •  •  •,  x^  in  terms  of  the  n  -|-  1  variables  x^, 


134  DIFFERENTIAL   CALCULUS 

Xp+i,  •  •  • ,  x„  +p,  and  the  results  may  be  substituted  in  F^.  It  remains  to  show  that 
Fj  =  0  is  soluble  for  x^.    Kow 

dF,       cF,       cF.  ex.,  cF,  cx„       ^    ^ 

~  =  ~  +  ^^  +  ---  +  -~^-^  =  J/Ji  ^  0.  (32') 

ax^       cx^       ex.-,  cJj  cXp  fjj 

For  the  derivatives  of  x^,  •  •  • ,  Xp  with  respect  to  x^  are  obtained  from  the  equations 

CX^  CX.-,  CX^  CXp  CX^  ZX-^  ex.,   cXj  CXp   CX^ 

resulting  from  the  differentiation  of  F.,  =  0,  •  •  •,  F^  =  0  with  respect  to  x^.  Tlie 
derivative  Zxjcx^  is  therefore  merely  J,/./j ,  and  hence  dFjdx^  —  J/J^  and  does 
n(jt  vanish.  The  equation  therefore  may  be  solved  for  x^  in  terms  of  J";, +  i.  •••, 
Xn  j^p.  and  this  result  may  be  substituted  in  the  solutions  above  found  for  x.,.  •  ■  ■ .  x^. 
Hence  the  equations  have  been  solved  for  x^,  x.,,  •  •  •,  Xp  in  terms  of  Xp  +i ,  •  •  • ,  x„  4-j, 
and  the  theorem  is  proved. 

For  the  second  theorem  the  procedure  is  analogous  to  that  previously  followed. 
If  there  is  a  relation  F(xiy.    •  •,  h^,)  =  0  between  the  p  functions 

"i  =  0i('''n  •  •  ■:  -^p)-  ■  ■  •'          ^P  =  <Pp{->'i-  •  •  •,  -fj), 
differentiation  with  respect  to   x^.  •  •  •,  ./Tp  gives  p  equations  fnun  which  the  deriva- 
tives of  F  bv  Hj.  •  •  •.  Up  mav  be  eliminated  and  ./(  — !^ •__j!\  _  q  i^ecomes  the  con- 

Ui-  ■•■,  .'•;./ 
dition  desired.  If  conversely  tliis  Jacobian  vanishes  identically  and  it  be  assumed 
that  one  of  the  derivatives  of  i;,-  by  X/,  say  cii-^/cx^.  does  not  vanish,  then  tlie  solution 
Xj  =  w(i<^,  X.,.  •  •  •.  x^,)  may  be  effected  and  the  result  may  be  substituted  in  u.,, 
•  •  •,  Up.  The  Jacobian  of  Ho.  •  •  •,  Up  with  respect  to  x,,.  •  •  ■.  Xp  will  then  turn  out 
to  be  ./  -^  (Hj/fXj  and  will  vanish  because  J  vanishes.  Now.  however,  only  _?>  —  1 
functions  are  involved,  and  lience  if  tlie  theorem  is  true  for  p  —  1  functions  it  nm.-it 
be  true  for  p  functions. 

EXERCISES 

1.  If   u  =  ax  +  hy  4-  c   and   v  =  a/x  +  h' ij  -\-  <■'  are  functionally  dependent,  the 
lines  u  =  0  and  v  =  0  are  parallel  ;  and  conversely. 

2.  Trove  X  +  y  +  z.  xy  +  yz  +  zx.  x-  +  y-  +  z-  functionally  dependent. 

3.  If  u  =  (IX  +  hy  +  €z  +  d,   V  =  a/x  +  h'y  +  c'z  +  (/'.    !'■  =  a"x  +  h"y  +  r"z  +  d" 
are  functionally  dependent,  tlie  planes  u  =  0.  i-  =  0,  v:  =  0  are  parallel  to  a  line. 

( V  t  d  F  c  F 

4.  In  wJiat  senses  are  —  and  -Jy,^  of  (24')  and  - — i  and  -  -  of  (82')  partial  or  total 

cy  '  '         '  (ix^  cXj 

derivatives  ■'    Are  not  the  two  sets  completely  analogous  ? 

5.  Given  (25).  suppose      ^        ^^0.    Solve  v  =  \p  and  lo  =  x  f'^'"  V  'I'l^^  ^i  substi- 


tute  in  H  =  (p.  and  prove  cu/cx  =  ./ 


6.  If  u  =  u  (x,  y),  V  =  f(x,  y).  and  x  =  x  (^.  17).  y  —  y(^.  rj),  prove 


State  the  extension  to  any  number  of  varialilcs.    llow  may  (27')  lie  used  to  prove 
(27) '.'    .Vgain  state  the  extension  to  any  nuuibcr  of  variaoies. 


PARTIAL   DIFFEREXTIATIOX ;   IMPLICIT 


135 


7.  Prove  dV  =  J 


X.  y. 


dudfdw  —  dwhdw 


u.  V.  iv\  . 
^—-  1.' 
X,  y,  zf 


is   tlie    element    of 


\u,  r,  10, 
volume  in  space  with  eurvilineai-  coordinates  »,  i-,  iv  —  consts. 

8.  In  what  parts  of  the  plane  can  u  =  x-  +  y-.  v  =  xy  not  be  used  as  curvi- 
linear coordinates  ?    Express  ds-  for  these  coordinates. 

9.  Prove  that  2  w  =  x-  —  y'-,  v  =  xy  U  a  confonnal  transformation. 


10.  Prove  that  x 


y 


is  a  conformal  transfoi'mation. 


11.  Define  conformal  transformation  in  space.    If  the  transformation 

X  =  au  +  bv  +  rii%         y  =  d'li  +  h'\:  +  c'v\         z  =  a"xi  +  h"x)  +  c"w 
is  conformal,  is  it  orthogonal  ?    See  Ex.  10  (f),  p.  100. 

12.  Show  that  the  areas  nf  the  trianules  whose  vertices  are 

(w,  r),  (w  4-  (Zu,  r),  (»,  i'  +  dn)     and     (m  +  dn.  x  +  dr).  (u  +  du,  t),  («,  v  +  dv) 
are  infinitesimals  of  the  same  order,  as  suuLrested  in  ^  '!o. 

13.  Would  the  condition  F=  0  in  (28)  mean  that  the  set  of  curves  u  =  const. 
were  perpendicular  to  tlie  set  v  =  const.  '.' 

14.  Express  E.  F.  (r  in  (28)  in  terms  of  the  derivatives  of  w,  v  by  x.  y. 

15.  If  X  =  0(.s.  t),  y  =  \p  (x.  t).  z  =  X  (•■*,  i)  'T-i'e  the  parametric  equations  of  a 
surface  (from  which  .s-.  t  could  be  eliminated  to  obtain  the  equation  between 
X,  ?/,  z),  show 


=  J 


X-  "/- 


.s.  t 


and  find 


cy 


65.  Envelopes  of  curves  and  surfaces.  Let  the  efitmtion  F(.<',  y,  a)  =  0 
be  coiisideved  ;is  it'prescutiii.n'  ;i  family  of  cui'ves  wliere  tlie  different 
curves  of  tlu?  family  are  obtained  by  assi.n'iiini;-  different  values  to  the 
]>arameter  a.    Such  families  are  ilhistrated  l)y 

(./■  —  ar  +  if  =  1     and     ax  +  ufa  =  1,  (33) 

wliieh  are  circles  of  unit  radius  centered  on  the  ./'-axis  and  lines  which 
cut  off  the  area  J- <r  from  the  first  (juadrant.  As  <x  changes,  the  circles 
remain  always  tangent  to  the  t\V(^  lines  7/  =  -j-  1  and 
the  point  of  tangency  traces  those  lines.  Again,  us  ^' 
a  changes,  the  lines  ('33)  I'emain  tangent  t(;  the  hyper- 
bola ./■//  =  /,•,  owing  to  the  pro})erty  of  the  hyperbola 
that  a  tangent  forms  a  triangle  of  c(jnstant  area  with 
the  asymptotes.  The  lines  y  =  +  1  are  called  the  — 
enreJopc  of  the  system  of  circles  and  the  liypcrlxjla 
,'/■//  =  /•  the  envelo])e  of  the  set  of  lines.  In  general,  If  there  is  a  rurre 
to  ii-li'ir],  f]ir  (■///■rt's  of  II  fdiiiih/  F(.i\  I/,  (i)  =  f)  lire  tiiniji'nf  nnrl  if  the, 
Jiniat   iif  til nijeilrij    ll rsrrilies    tJitit    nirre    us    a    mrirs,    the    rurre    is   ciilti-d 


186  DIFFERENTIAL   CALCULUS 

ilic  enreh>j/e  (or  part  of  the  envelope  if  there  are  several  such  curves) 
of  the  f<i mil ij  F(^x,  y,  a)  =  0.  Thus  any  curve  may  be  regarded  as  the 
envelope  of  its  tangents  or  as  the  envelo})e  of  its  circles  of  curvature. 

To  find  the  ecpiations  of  the  envelope  note  that  by  definition  the 
enveloping  curves  of  the  family  F{/,  //,  a)  =  0  are  tangent  to  the  envelope 
and  tliat  tlui  j)oint  of  tangency  moves  along  the  envelo])e  as  a  varies. 
Th(!  ecpiation  of  the  envehjpe  may  therefore  be  written 

:,'  =  ^{(t),  y  —  i{/(o:)     with     Fief),  if/,  a)  =  0,  (;>4) 

where  the  lii-st  ecpiations  express  the  dependence  of  the  ])oints  on  tlie 
envelope  uj)on  tlie  parameter  a  and  the  last  ecj^uation  states  that  eaeli 
point  of  the  envelope  lies  also  on  some  curve  of  the  family  F(.r,  y,  a)  =  0. 
Differentiate  (84)  with  respect  to  a.    Then 

F:.cj>'(a)  +  i-;>'(  '0  +  f;  =  0.  (35) 

Xow  if  the  ]ioint  of  contact  of  tlie  enveloi)e  with  tlie  curve  i^  =  0  is  an 
ordinary  point  of  that  curve,  the  tangent  to  the  cui've  is 

F;(.r  -  ./■  j  +  F;(y  -  yj  =  0  ;      and      F;,j,'  +  i-^^'  =  0, 

since  the  tangent  direction  dy  :  dj'  =  i//'  :  <^'  along  the  envelope  is  hx 
definition  identical  A\itli  that  along  the  envelo]iing  curve;  and  if  the 
point  of  contact  is  a  singular  point  for  the  enveloping  curve,  F'^.  =  7-^  =  0. 
Hence  in  either  case  F^^  =  0. 

Thus  _/'o/' //o/'/iAs-  oil  fill'  inircliijic  the  fico  i-tjinifion.^ 

Fi,;  y,  a)  =  0,  F^-';  y,  a)  =  0  (36) 

(ifp,  Sdt'islicil  ond  flic  ciiiuif'ion  <>f  flw  mciJoiii'  of  fJic  fn  mil  //  F  =:  0  may 
hf  found  liij  siiJi-iiuj  (.')<")j  fu  fi  ml  ilic  pa  re  tin'fric  nj  iKifimis  ./■  =  (f)(n). 
y^i^(a)  of  flic,  ('iiri'h)iii',  or  In/  cliiiiuiot'iiKj  a  hrfirccn  (•")(>)  in  find  1hi' 
cijinif'Kin  of  flic  cnrclojic  in  flic  firm  <i>(.r.  y)  =  0.  It  should  be  remai-ked 
that  the  lonis  found  by  this  ])Voct'SS  iiiay  contain  other  curves  than  the 
envelope,  fm-  instanre  if  the  curves  of  the  family  F  =  0  have  singular 
])(Mnts  and  if  ./•  =  4>('t),  .'/ =  (//(a)  be  the  locus  of  the  singular  ])oints 
as  a  varies.  e(pialions  CM).  (.">.'))  still  hold  and  hence  (.')('))  also.  The 
rule  for  finding  the  envelojie  tliend'ore  finds  also  the  locus  of  singular 
])oints.  ()thei'  exti-ani'ous  factors  niav  also  l)e  introduced  in  perfoi-niing 
the  (diminalion.  Jt  is  there I'oi-e  ini])orlant  to  test  graphically  or  analvt- 
icallv  the  solution  obtained  by  applying  the  rule. 

As  a  tirst  cxaiiijilc  let  tlie  cuvclu^jc  of  (./•  —  n)-  +  //-  ~  1  lie  feniid. 

Fi.r.  II.  a)-  is  -(11-  +  //■-  -1=0.  Fj  -  -  2  (./•  -  ,( )  =  0. 

The  cliuiiiiatieii  of  (X  from  tlicsc  (■qnations  L;i\('s  //-—I  =0  and  tlic  solution 
for  It    'A\v>  X  ■'  ix.  a  ^~  \   1.    'riie   loci    indicatcil  as  cmclopcs   ai'c    ij  =  ±  1.    It  Is 


PARTIAL   DIFFEREXTIATIOX :   lAIPLICIT 


137 


geometrically  evident  that  tliese  are  really  envelopes  and  not  extraneous  factors. 
But  as  a  second  example  consider  ax  +  y/ct  =  1.    Here 


i'ir,  y, 


o:^.  +  y/et  —1  =  0,         F'^  —  x  —  y  /  a-  =  0. 


The  solution  is  y  =  a/2,  x  =  I /'2a,  which  gives  xy  =  \.  This  is  the  envelope ;  it  could 
not  be  a  locus  of  singular  points  of  i^  =;  0  as  there  are  none.  Suppose  the  elimina- 
tion of  a  be  made  \)y  Sylvester's  method  as 


y/a-  +  O/ir  +  .e -(- Oa  =  0 

0/rt-  —y/a  +  0  +  .'vr  =  0 

y/a-  —  \/a  +  x  +  ()(t:  =  0 

0/a-  +///<(  -  1  +  J-cr  =  0 


and 


y 

0 

x     0 

0 

—  // 

0     X 

y 

- 1 

X     0 

0 

y 

-   ]       X 

=  0 


the  reduction  (jf  the  detei'minant  gives  xy{ixy  —  1)  =:  0  as  the  eliminant,  and  con- 
tains not  only  the  envelope  ixy  =  I,  but  the  factors  x  =  0  and  y  =  0  which  are 
obviously  extraneous. 

As  a  third  problem  find  the  envelope  of  a  line  of  which  the  length  intercepted 
between  the  axes  is  constant.    The  necessary  ecpiations  are 


it-  -f-  13-  -  K-, 


^  (la  -f  ■'   (7/3 

:-  /3- 


0. 


ada  +  jid^  =  0. 


Two  parameters  a,  j3  connected  bj'  a  relation  have  been  intro(bTced  ;  both  ecpations 
have  been  differentiated  totally  with  respect  to  the  parameter.s  ;  and  the  problem 
is  to  elinunate  a.  /3,  da,  dj3  from  the  equations.  In  this  case  it  is  simpler  to  cai'ry 
both  parameters  than  to  introduce  the  radicals  which  would  be  recjuired  if  oidy 
one  parameter  wi're  used.  The  elimination  of  do:,  d(i  from  tlie  last  two  equations 
gives  X  :  y  =  a' :  /S'*  or  Vx  :  V^/  =  a  :  /3.    From  this  and  the  lirst  e(juation, 

1  _  1  1  _  1 

<^^         j:j  {x~i   Jf-  ys) 


/3 


y  '■■■.  (xTi  +  ?/3 


0 


and  hence 


+  y/S  =  Zv'J. 


66.   Consider  two  iieig-]il)oriiig  curves  of  F(.r,  //,  a)  ~  0.     Let  (.r^^,  y  ) 
be  an  ordiiiary  })oiiit  of  a  =  a^^  and  (.r^^  +  (/./■,  //.  -|-  <///)  of  a'^^  -(-  r/a-.    Then 

=  F:>/.r  +  F;^,/;/  +  F:,]a  =  0  {?u) 

holds  ex('e])t  for  infinitesimals  of  lii<;lH'r  order.    Tlie  distance  from  tlie, 
point  on  a^  +  "'''<-'  to  the  tani^-ent  to  a^^  at  (x^^,  i/^^  is 


±  Vi-'/  -f  /■;;-      Vy-::-  +  f/ 


(In. 


(38) 


except  for  infinitt>simals  of  liii^'lier  order.  Tliis  distance  is  of  the  fii'st 
order  witli  c/a'.  and  tlie  normal  derivati\-e  dajdn  of  §  4(S  is  finite  excejtt 
when  F[^  =  0.  T'he  distance  is  of  liigher  order  than  da,  and  (/a/(/)i  is 
infinites  or  <ln jdn  is  zero  when  F[^  =  0.  It  a])])ears  therefore  that  flw 
cnri'liijip  !s  the  locii.^  nf  jiDinfs  at  irhii'li  flic  (llxfuncc  hcticrcn  firn  nchjji- 
borlnij  cKrrcs  is  of  Ii  ii/]icr  order  tlinii  da.  This  is  also  apparent  geomet- 
rically fi-om  tlie  i'act  that  the  distance  from  a  ]ioint  on  a  cnrve  to  the 


138  DIFFERENTIAL  CALCULUS 

tangent  to  the  curve  at  a  neighboring  ])oint  is  of  higher  order  (§  36). 
Hinguhir  points  have  l)een  ruled  out  because  (38)  l)ecomes  indeternii- 
iiate.  In  general  the  locus  of  singular  points  is  not  tangent  to  the 
curves  of  the  family  and  is  not  an  envelope  but  an  extraneous  factor ; 
in  exceptional  cases  this  locus  is  an  envelope. 

If  two  neighboring  curves  F(.r,  //,  a)  =  0,  F(.r,  y,  a  -\-  A«)  =  0  inter- 
sect, their  point  of  intersection  satisfies  both  of  the  equations,  and  hence 
also  the  equation 

—  lF{x,  u,  a  +  \a)  -  F{.r,  //,  «)]  =  F'^  (,',  ,/,  a  +  6\a)  =  0. 

If  the  limit  be  taken  for  A«  =  0,  the  limiting  position  of  the  intersec- 
tion satisfies  F^  =  0  and  hence  may  lie  on  the  envelope,  and  will  lie  on 
the  envelope  if  the  common  point  of  intersection  is  remote  from  singular 
points  of  the  curves  F(x,  t/,  a)  =  0.  This  idea  of  an  rnrr/ape  as  f/te 
Ill/tit  of  points  ill  irhlrh  neUjlihor'tng  rinu-es  of  flie  funtili/  Infi'i'st'rf  is 
valuable.  It  is  sometimes  taken  as  the  definition  of  the  envelo})e.  But, 
unless  imaginary  points  of  intersection  are  considered,  it  is  an  inade- 
quate definition  ;  for  otherwise  //  =  (,/'  —  af  would  have  no  envelope 
according  to  the  definition  (whereas  i/  =  0  is  obviously  an  envelope)  and 
a  curve  could  not  be  regarded  as  the  envelope  of  its  osculating  circles. 

Care  must  be  used  in  applyiiii,''  the  rule  for  tiiiding  an  envelope.  Otherwise  not 
only  may  extraneous  solutions  be  mistaken  for  the  envelope,  but  the  envelope  may 
be  missed  entirely.    Consider 

y  —  sin  a/  =  0     or     a  —  /-i  sin-i  y  =  0.  (39) 

where  the  second  form  is  obtained  by  solution  and  eontains  a  multiple  valued 
function.  These  two  families  of  curves  are  identical,  and  it  is  iieometrii-ally  clear 
that  they  have  an  envelope,  namely  y  =  ±  1 .  This  is  precisely  what  woidd  l)e 
found  on  aiiplyin,ii-  the  I'ule  to  the  first  of  (3!>)  ;  but  if  the  ride  be  applied  to  the 
second  of  (3l»),  it  is  seen  that  7^  =  1,  which  does  not  vanish  and  hence  indicates  no 
envelope.  The  whole  matter  slioidd  be  examined  carefully  in  the  liuiit  of  ini])licit 
functions. 

IIeiic(^  let  F(s,  y.  a)  =  0  be  a  contiiuious  single  valued  function  of  the  three 
\ai'iables  (,c.  y.  a)  and  let  its  <lei-ivatives  F'^..  F',^.  F^  exist  and  be  continudus.  Con- 
sider the  l)ehavior  of  the  curves  of  the  family  near  a  ixiint  (/,,.  ?/,,)  of  the  curve  for 
a  —  <i-|,  provided  that  (,/•,,.  ?/,,)  is  an  ordinary  (uonsinuular)  point  of  the  curve  and 
that  the  derivative  F'^(x^^.  //,,.  (t-„)  does  not  vanish.  As  F[^  ^t  0  and  eitiier  F\.  #  0 
or  7-',^  ^  0  for  {.r^^,  t/^.  (i,,),  it  is  jtossible  to  surround  (,f,,,  ?/,,)  with  a  region  so  small 
that  F(.r.  y.  a)  :=  0  may  l)e  sohcd  for  a  =  f(,r,  y)  which  will  be  sini;le  \alued  and 
differentialile :  and  tin-  I'ciiiou  may  further  be  taken  so  small  that  F'^.  or  /-'J  irmains 
different  from  0  tlirouuhout  the  reuion.  Then  throui:h  every  jioint  of  tlic  re,i;ion 
there  is  one  and  oidy  one  curve  a  =f(s.  y)  and  the  curves  have  no  siniiular  points 
within  the  reiiion.  In  iiarticulai-  no  two  curves  of  the  family  can  be  tangent  to 
each  other  ^vithill  the  i'cL;ion. 


PARTIAL  DIFFEKENTIATIOX;   I.AIIM.KJIT  189 

Furthci-more,  in  such  a  region  there  is  no  envelope.  For  let  any  curve  which 
traverses  the  region  be  x  =  <^  {t),  y  —  \p  (t).    Then 

a  (t)  =  /(0  (t),  V  (0),         «'(0  =  />'(<)  +  />'(0- 

Along  any  curve  a  =/(x,  y)  the  equation  f/lx  +//?//  =  0  holds,  and  if  x  =  4>{t), 
y  =  \l/{t)  be  tangent  to  this  curve,  dy  —  dx  =  \j/'  -.  (p'  and  cx'{t)  =  0  or  a  =  const. 
Hence  the  only  curve  which  has  at  each  point  the  direction  of  the  curve  of  the 
family  through  that  point  is  a  curve  which  coincides  throughout  with  some  curve 
of  the  family  and  is  tangent  to  no  other  member  of  the  family.  Hence  there  is  no 
envelope.  The  result  is  that  an  envelope  can  be  x)resent  only  when  F^  =  0  or  when 
F'^  =  jp,^  =  0,  and  this  latter  case  has  been  seen  to  be  included  in  the  condition 
F^  =  0.  If  F{x,  ?/,  a)  were  not  single  valued  but  the  branches  were  separalile,  the 
same  conclusion  would  hokl.  Hence  in  ease  F(x,  ?/,  a)  is  not  single  valued  the  loci 
over  which  two  or  more  values  become  inseparable  must  be  added  to  those  over 
which  F^  =  0  in  order  to  insure  that  all  the  loci  which  may  be  envelopes  are  taken 
into  account. 

67.  The  preceding  considerations  apply  with  so  little  change  to  other 
cases  of  envelopes  that  the  facts  Avill  niei'ely  be  stated  Avithout  proof. 
Consider  a  family  of  siu'faces  F(.i',  y,  ,^',  a,  (i)  =  0  depending  on  two 
parameters.  The  envehjpe  may  be  defined  by  the  property  of  tangency 
as  in  §  65;  and  tlia  condlf  ions  for  an  enrclopa  waidd  he 

F(x,  >/,  z,  a,  /3)  =  0,  F:  =  0,  7-',  =  0.  (40) 

These  three  equations  may  be  solved  to  express  the  envelope  as 

x  =  cf,  (a,  13),  ii  =  ip  (a,  j3),  ::  =  x  (o:,  /3) 

parametrically  in  tcrins  of  <i,  (3;  or  the  two  parauu'ters  may  be  elimi- 
nated and  the  envelope  may  be  found  as  <!»(.'•,  //,  .'.)  =  0.  In  any  case 
extraneous  loci  niay  be  introduced  and  the  results  of  the  Avork  should 
therefore  be  tested,  which  generally  may  be  done  at  sight. 

It  is  also  possible  to  determine  the  distance  from  the  tangent  plane 
of  one  surface  to  th(>  neighboring  surfaces  as 

^K'  +  K  +  K'         -^K'  +  K  +  K' 

and  to  define  the  envelope  as  the  locus  of  j)oints  su(^h  that  this  distance 
is  of  higher  order  than  \i/a\  +  \dft\.  The  equations  (40)  would  then  also 
follow.  This  definition  would  a})}ily  only  to  ordinary  points  of  tlie  sur- 
faces of  the  family,  tliat  is,  to  points  for  Avhich  not  all  tlie  derivatives 
F',.,  F',i,  F^  vanish.  lUit  as  the  elimination  of  a,  f3  from  (40)  would  give 
an  ecpuition  Avliich  included  the  loci  of  these  singular  points,  there 
would  l)e  no  danger  of  losing  such  loci  in  the  rare  instances  Avhere  they, 
too,  happened  to  be  tangent  to  the  surfaces  of  the  family. 


140  DIFFEKEXTIAL   CALCULUS 

The  application  of  implicit  functions  as  in  §  00  could  also  be  made  in  tliis  case 
and  would  show  that  no  envelope  could  exist  in  regions  where  no  singular  points 
occurred  and  where  either  F^  or  F'^  failed  to  vanish.  This  work  could  be  based 
either  on  the  first  definition  involving  tangency  directlj-  or  on  the  second  definition 
which  involves  tangency  indirectly  in  the  statenients  concerning  infinitesimals  of 
higher  order.  It  may  be  ailded  that  if  F(x,  ?/.  2,  a-,  (3)  ==  0  were  not  single  valued, 
the  surfaces  over  which  two  values  of  tlie  function  become  inseparable  should  be 
added  as  possible  envelopes. 

A  family  of  surfaces  F(x,  //,  ,--,  a)  =  0  dependiuf.,^  011  a  single  param- 
eter may  have  an  envelope,  and  tlm  encclope  is  fmoid  from 

F(.r,  v/,  .V,  a)  =  0,  F^Cr,  >/,  ;:,  a)  =  0  (42) 

hj  the  elimination  of  the  single  i)arameter.  Tlie  details  of  the  deduction 
of  the  rule  will  l)e  omitted.  If  two  neighboring  surfaces  intersect,  the 
limiting  position  of  the  curve  of  intersection  lies  on  the  envelope  and 
the  envelope  is  the  surface  generated  by  this  cur\-e  as  a  varies.  The 
surfaces  of  the  family  touch  the  envelope  not  at  a  i)oint  merely  l)ut 
along  these  curves.  The  curves  are  called  clini-di-fin-lstlrs  of  the  family. 
In  the  case  where  consecutive  surfat-es  of  the  family  do  not  intersect 
in  a  real  curve  it  is  necessary  to  fall  l)ack  on  the  conception  of  imagi- 
nai'ies  or  on  the  definition  of  an  enveh)pe  in  terms  of  tangency  or 
infinitesimals ;  the  characteristic  curves  are  still  the  curves  along 
which  the  surfaces  of  the  family  are  in  contact  with  the  envelo})e  and 
along  which  two  consecutive  surfaces  of  the  family  are  distant  from 
each  other  l)y  an  infinitesimal  of  higher  order  than  da. 

A  particular  case  of  importance  is  the  envelope  of  a  i)lane  which 
depends  on  one  parameter.    The  equations  (42)  are  then 

Ax  +  ]'.;/  -h  ^.'.v  +  n  --=  0,  J'.r  4-  />"//  4-  (_"::  +  D'  =  0,        (43) 

where  .1,  B,  C,  D  arc  functions  of  tlie  parameter  and  differciutr'ation 
with  respect  to  it  is  denoted  by  accents.  The  case  where  the  plaiu' 
moves  parallel  to  itself  or  turns  about  a  line  may  lie  excluded  as  trivial. 
As  the  intersection  of  two  planes  is  a  line,  tlie  characteristics  of  tin' 
system  are  straight  lines,  the  envelope  is  a  i-iilrd  surfiny,  and  ca  piaur 
tangent  to  tJui  niir/'icn  at  one -point  of  tlie  /ini's  is  funijent  to  the  surfdcf 
tlirour/hout  the  vJioIe  extent  of  the  tine.  Cones  and  cylindei'S  are  exam- 
ples of  this  sort  of  surface.  Another  exam}»le  is  the  surface  enveloj)ed 
In'  the  osculating  ])lanes  of  a  curve  in  space ;  for  the  oscidating  plane 
depends  on  only  one  parameter.  As  the  osctdating  plane  (§  41)  mav  be 
regarded  as  passing  tlirough  three  consecutive  points  of  the  cu)-ve,  two 
consecutive  osculating  planes  may  l)e  considei'ed  as  liaving  two  consecu- 
tive points  of  the  curve  m  common  and  hence  the  characteristics  are 


PARTIAL  DTFFEKENTIATION  ;   IMPLICIT  141 

the  tangent  lines  to  the  eurve.    Suvi'aees  which  are  the  envelopes  of  a 
plane  which  dej)entls  on  a  single  })aranieter  are  called  th-n-hqjohh:  mirfncca. 
A  family  of  curves  dependent  on  two  })aranieters  as 

1-i.r,  :j,  ::,  a,  j3)  =  0,  G  (x,  //.  ,-:,  a,  (3)  =  0  (44) 

is  called  a  coiuir^nnvc  nf  cnrrfs.  The  curves  niay  have  an  envelope,  that 
is,  there  niav  l»e  a  surface  to  which  the  curves  ai'e  tangent  and  which 
may  be  regarded  as  the  locus  of  their  })oints  of  tangency.  The  envelope 
is  obtained  by  eliminating  a.  /5  from  the  ecpiations 

F  =  0,         <;  =  0,         F'ji,  -  7-0-  =  0.  (45) 

To  see  this,  sup})ose  that  the  third  condition  is  not  fulfilled.  The  equa- 
tions (44)  may  then  be  solved  as  a  =  /-'(■'',  //,  '-'),  /3  =  ,v(.'',  //.  .-).  Leason- 
ing  like  that  of  §  6(>  now  shows  that  there  cannot  possibly  lie  an 
envelope  in  the  region  for  which  the  solution  is  valid.  It  may  therefore 
l)e  inferred  that  the  only  possibilities  for  an  envelope  are  contained  in 
die  equations  (4o).  As  various  extraneous  loci  might  be  introduced  in 
the  elimination  of  a.  /3  from  (4."ij  and  as  tlu^  solutions  should  therefore 
be  tested  individually,  it  is  hardly  necessary  to  examine  the  general 
question  further.  The  envelope  of  a  congruence,  of  curves  is  called  the 
focdj  siirfdcc  of  the  congruence  and  the  points  of  contact  of  the  curves 
with  the  envelope  are  called  i\w  /'><■"!  poinfa  on  the  curves. 

EXERCISES 

1.  Find  the  oiivolopes  of  tlicse  families  uf  cur\c's.    In  each  case  test  the  answer 
or  its  individual  factors  and  check  the  results  l>v  a  sketch  : 

{a)   >j  =  2  a.c  +  a-4,  (,i)  y-  =  <t  (,r  -  a),         (y)  ;/  =  a.r  +  k/a, 

(5)  a{y  +  af  -  x^,         (e)  y  -  a{.c  +  a)-,         (f)  //-  =  a{.i:  -  (if. 

2.  Find  the  envelope  (if  the  ellipses  x'-/<i'-  +  y- /^r  =  1  under  the  eunditidii  that 
(a)  the  sum  of  tlie  axes  is  constant  or  (^^)  the  area  is  constant. 

3.  Find  the  enveh'pe  of  the  circles  whose  center  is  on  a  j:iven  jjarabola  and 
which  pass  through  the  vertex  of  the  parabola. 

4.  Circles  pjass  throuyh  the  origin  and  have  their  centers  on  j-  —  //-  =  r'-.    Find 
their  envelope,  xl?i.s.  A  leinniscate. 

5.  Find  the  envelopes  in  these  cases  : 

(a)  X,  +  xya  —  sin-  ^  x.y.         (3)  x+  a  ■=  vers-  i  y  +  V2  y  —  y-, 
{'/)  y  +  a  =  Vl-  l/x. 

6.  Find  the  envelopes  in  these  cases  : 
{a)  ax  +  py+  a^z  =  1.         (/?)  i^  +  ^  -f 


/:*        1  —  «  — 

(7)  —  +  -,  +  ~  =  1  ^vith  aiiy  =  k^. 
a-      /3-      7- 

7.  Find  the  envelopes  in  Ex.  (J  [a).  {[3)  if  a  =  (3  or  if  cr  =  — 


142  DIFFEREXTIAL   CALCULUS 

8.  Prove  that  the  envelope  of  F{x.  y.  z,  a)  =  0  is  tangent  to  the  surface  along 
the  whole  characteristic  by  showing  that  the  normal  to  F(x,  y,  z.  a)  =  0  and  to  the 
elimiiiant  of  i^"  =  0,  F^  =  0  are  the  same,  namely 

F;  :  F' :  F:     and     F^  +  F'^  ' "  :  F'  +  F^~  :  Fl  +  F;^  —  , 
■^      '■  cx,  cy       ~  Zz 

where  a:(x,  ?/,  z)  is  the  function  obtained  by  solving  F„'  =  0.   Consider  the  problem 
also  from  the  point  of  view  of  infinitesimals  and  the  normal  derivative. 

9.  If  there  is  a  curve  x  =  (p{a),  y  =  f{a),  z  —  xi^^)  tangent  to  the  curves  of 
the  family  defined  by  F{x,  y,  z,  a)  =  0,  G  (x,  y,  z.  a)  —  0  in  space,  then  that  curve 
is  called  the  envelope  of  the  family.  Show,  by  the  same  reasoning  as  in  §  65  for 
the  case  nf  the  x'laiie,  that  the  four  conditions  F  =  0,  ft  =  0.  F^  =  0,  G^  =  0  must 
be  satisfied  for  an  envelope  ;  and  lience  infer  that  ordinarily  a  family  of  curves  in 
space  dependent  on  a  single  parameter  has  no  envelijpe. 

10.  Show  that  the  family  F(x,  y,  z,  a)  =  0,  F^[{x,  y.  z.  a)  =  0  oi  curves  which 
are  the  characteristics  of  a  family  of  surfaces  lias  in  general  an  envelope  given  by 
the  three  equations  7^"'  =  0,  Fa  =  0,  F^'^  =  0. 

11.  Derive  the  condition  (45)  for  the  envelope  of  a  two-parametered  family  of 
curves  from  the  idea  of  tangency,  as  in  the  case  of  one  parameter. 

12.  Find  the  envelope  of  the  normals  to  a  plane  curve  y  =/(x)  and  show  that 
the  envelope  is  the  locus  of  the  center  of  curvature. 

13.  The  locus  of  Ex.  12  is  called  the  eiolute  of  the  curve  y  =/(x).  In  the.se  cases 
find  the  evolute  as  an  envelope  : 

[a)  y  =  X-,  (/3)  X  =  a  sin  t.  y  =  b  cos  t,  (y)  2  xy  =  n-, 

(5)  y-  =  2  mx,         (e)  x  =  a{d  —  sin  0).  y  =  r/  (1  —  cosi9),  (f)  y  =  coshx. 

14.  Given  a  surface  z  =/(x,  y).  Construct  the  family  of  normal  lines  and  find 
their  envelope. 

15.  If  rays  of  liglit  issuing  from  a  point  in  a  plane  are  reflected  from  a  curve  in 
the  plane,  the  angle  <if  reflection  Ix'ing  ecpial  to  tlu'  angle  of  incidence,  the  envelope 
fif  the  retiected  rays  is  called  the  rdiistlr  of  the  curve  witli  respect  to  the  point. 
Show  that  the  caustic  of  a  circle  with  respect  to  a  point  on  its  circumference  is  a 
cardioid. 

16.  The  curve  which  is  the  envelope  of  the  cliaracteristie  lines,  that  is.  f^if  the 
rulings,  on  the  developable  surface  (4o)  is  called  the  cuspidal  edye  of  the  surface. 
Show  that  the  eiiuations  of  this  curve  may  Ije  found  parametrically  in  terms  of  tlie 
parameter  of  (43)  by  solving  simultaneously 

.Ix  +  By  +  Cz  +  D  =  0.  A'x  +  B'y  +  ("z  +  T)'  =  0.  .4"x  +  B"y  +  C"z  +  D"  =  0 

for  X,  y.  z.    Consider  the  exceptional  leases  of  cones  and  cylinders. 

17.  The  term  "  developable  "'  signifies  that  a  ilrrchipdlih;  .■surface  riKiy  he  dncloped 
or  majjped  on  a  jAane  in  Hudt  a  iray  tluit  lenr/tJ/s  of  nrrs  nn  tin:  sur/urr  hvrmnv  equal 
lenytJis  in  Ihc  plane,  that  is.  the  map  may  be  made  without  distortion  of  sizt-  or 
shape.  In  the  case  of  cones  or  cylinders  this  map  may  V)e  made  by  slitting  the  cone 
or  cylinder  along  an  element  and  rolling  it  out  upon  a  plane.  What  is  the  analytic 
statement  in  this  case?  In  the  case  of  any  developabh*  surface  with  a  cuspidal 
edge,  the  developable  surface  being  the  hx'us  of  all  tangents  to  the  cuspiihil  edge. 


PARTIAL   ])1FFEKEXTIATI0X;   I3I1*LI(J1T  143 

the  length  of  arc  upon  the  surface  may  be  written  as  do--  =  {dt  +  da)"  +  t-ds~/E-, 
where  s  denotes  arc  measured  along  the  cuspidal  edge  and  t  denotes  distance  along 
the  tangent  line.  This  form  of  dff-  may  be  obtained  geometrically  by  infinitesimal 
analysis  or  analytically  from  the  equations 

X  =/(.s)  +  (r(.s),      y  =  rj{,)  +  tr/{.),      z  =  h  (.s)  +  th'{s) 

of  the  developable  surface  of  which  x  =/(.s),  y  =  ^(-s),  z  =  Ii{»)  is  the  cuspidal  edge. 
It  is  thus  seen  that  da-  is  the  same  at  corresponding  points  of  all  developable  sur- 
faces for  whicli  the  radius  of  curvature  R  of  the  cuspidal  edge  is  the  same  function 
of  .s-  without  regard  to  the  t<n\sion  ;  in  particular  the  torsion  may  be  zero  ami  the 
developable  maj'  reduce  to  a  plane. 

18.  Let  the  line  x  =  az  +  b.  y  =  rz  +  d  depend  on  one  parameter  so  as  to  gen- 
erate a  ruled  surface.  By  identifying  this  form  of  the  line  with  (43)  obtain  by 
substitutiiin  the  conditions 

Aa  +  Jk  +  ('  =  0,     A  '<i  +  B'c  -j-  C"  =  0  An'  -\-  Br'  =  0        _    J 

Ah  +  Bd  +  IJ=  0.     A'h  +  B'd  +  D' =  0     "'"     Ab'  +  Bd'  =  0     *^'^"    16'  d'\ 

as  the  condition  tiiat  the  line  generates  a  develoijable  siu'face. 


=  0 


68.  More  differential  geometry.    The  representation 

F(,r,  ;/,z)=0,      or      z=f(.r^,/)  (46) 

or  ,/•  =  (fi(>',  '•),  //  =  ^(",  '■),  ;:  =  -^(ii,  r) 

of  u  stirfaee  nuiy  be  taken  in  the  luisolved,  the  solved,  oi-  tlic  ])aranietric 
form.  The  panuuetric  form  is  e(|tiivalent  to  the  solved  form  pr(jvided 
V.  r  he  taken  as  ./•,  //.    The  notation 

C-:  C-:  c'-r:  C'r:  _  C'z 

ex  Cjl  C.r-  CXCiJ  Cif 

is  adojited  for  the  dci'i\'atives  of  ;.■  with  ivsjiect  U)  x  and  //.  The  applica- 
tion of  Taylo]''s  Formula  to  the  solved  form  gives 

A^  =  ,,h  +  ,,h  -f  \  (rh-  +  2shk  +  fir)  -f  •  •  •  (47) 

with  //  =  \.i\  /,•  =  A//.  Tlie  linear  terms  y///  +  '//>■  constitute  the  differ- 
ential '/,-;  iuid  rejjresent  that  part  (jf  tlie  increment  of  x  which  wotild  l)e 
ol)tained  liy  replacing  the  surface  by  its  tangent  ])lane.  Apai't  from 
intinitesinutls  of  the  third  ordei-.  the  distance  from  the  tangent  [ilane  \\\) 
or  down  to  the  surface  along  a  parallel  to  the  ,-;-axis  is  given  \)\  the 
(|tiadratie  terms  \{r]r  4-  2s]il:  -\-  fir). 

Hence  if  the  quadratic  terms  at  any  ])oint  are  a  positive  definite  foiau 
(§  55).  the  surface  lies  above  its  tangent  plane  and  is  concave  \\\i :  but 
if  the  f(ji-m  is  negative  definite,  the  surface  lies  Ixdow  its  tangent  }»lane 
and  is  concave  down  or  c(jnvex  up.  If  the  form  is  indehnite  but  not 
singular,  the  surfai-e  lies  partly  aljove  and  partly  below  its  tangent 
}ilam:'  and  may  be  called  concavo-convex,  that  is,  it  is  saddle-sha]>e(l.  If 
the  form  is  singular  nothing  can  b;-  detinitelv  stated.    These  statements 


144  DIFFERENTIAL   CALCULUS 

are  merely  generalizations  of  those?  of  §  5o  made  for  the  case  where  the 
tangent  plane  is  parallel  to  the  .ry-plane.  It  will  be  assumed  in  the 
further  work  of  these  articles  that  at  least  one  of  the  derivatives  r,  .s,  t 
is  not  0. 

To  examine  more  closely  the  behavior  of  a  surface  in  the  vicdnity  of 
a  particular  point  upon  it,  let  the  .r//-plane  be  taken  in  coincidence  with 
the  tangent  plane  at  the  point  and  let  the  }K)int  be  taken  as  origin. 
Then  Maclaurin's  Formula  is  available. 


;:;  =  ^(rx'  +  2  ,svr y  +  fi/~)  +  terms  of  higher  order 
—  2"  P~(''  ^•^^'  ^  +  2  .s-  sin  9  cos  6  -}-  t  sin-  6)  -\-  higher  terms, 
where  (p,  6)  are  polar  coordinates  in  the  .ry-plane.    Then 


—  =  /'  (;os-  ^  +  2  .s  sin  0  cos  6  -\-  t  sin'-  0  =  —-^, 
R  dp' 


(48) 


(40) 


is  the  curvature  of  a  normal  section  of  the  surface.  The  sum  of  the 
curvatures  in  two  noi'mal  sections  which  are  in  })erpendicular  planes 
may  be  obtained  by  giving  0  the  values  6  and  6  + -^  tt.  This  sum 
reduces  to  r  -\-  t  and  is  therefore  independent  of  9. 

As  the  sum  of  the  ('urvatures  in  two  ptu'pendieular  normal  planes  is 
constant,  the  maxinnim  and  mininuim  values  of  the  curvature  will  b(i 
found  in  perpendicular  })lanes.  Tliese  values  of  the  curvature  are  called 
the  principal  vffJKcs  and  their  reci])rocals  are  the  p))'i7icip(il  rajHi  of 
curvature  and  the  sections  in  wliich  they  lie  are  the  prinviiKil  scrtians. 
If  s  =  0,  the  principal  sections  are  ^  =  0  and  9  =  ^,77;  and  conversely 
if  the  axes  of  x  and  //  had  been  chosen  in  the  tangent  })lane  so  as  to  l)e 
tangent  to  tlie  principal  sections,  tlie  derivative  .s  would  have  vanished. 
The  equation  of  the  surface  would  then  have  taken  the  simple  form 

?^  =  1  (rx'^  +  ftp)  +  higher  terms.  (.■>()) 

The  principal  curvatures  would  be  merely  r  and  f,  and  the  cur\'ature 
in  any  normal  section  would  have  had  the  form 

1       cos-  9       sin-  ^  .,  „  .    ,  ^ 

—  =  — ; 1 ^ —  =  ;•  cos-  9  +  t  sin-  9. 

If  the  two  ])rinc;i])al  curvatures  have  0])posite  signs,  that  is,  if  the 
signs  of  ;•  and  t  in  (50)  are  0})posite,  tlie  sui'face  is  saddle-shaped. 
There  are  then  two  directions  for  Avhich  the  curvature  of  a  normal  sec- 
tion vanishes,  namely  the  directions  of  the  lines 

$  =  ±  taii^i  V—  /.',  /y.'j     or      Vpl  ./■  =±  V|  /*  I  //. 

These  nvo,  calliMl  tlie  axi/nipfufir  (lirccfions.  Along  these  directions  the 
surface  departs  from  its  ta,ngeiit  plane  liy  iniinitesimals  of  i\w.  third 


TAKTIAL   DIFFEIiEXTIATIOX;   l.^tPLlClT  145 

ordei-,  or  higlier  order.  If  ii  curve  is  drawn  on  a  surface  so  that  at  each 
point  of  tlie  curve  the  tangent  to  the  curve  is  along  one  of  the  as^-ni})- 
totic  directions,  the  curve  is  called  an  ai<ijiuptntlc  (■iirre  or  line  of  the 
surface.  As  the  surface  departs  from  its  tangent  plane  by  inhnitesimals 
of  higher  order  than  the  second  along  an  asymptotic  line,  the  tangent 
I)lane  to  a  surface  at  any  point  of  an  asymptotic  line  must  be  the  oscu- 
lating plane  of  the  asymptotic  line. 

The  character  of  a  }>oint  upon  a  sui-face  is  indicated  by  the  Dupln 
lmllc((trix  of  the  })oint.    The  indicatrix  is  the  conic 

;^  +  f  =  l,  ^L::  =  l{i:.-  +  t!r),  (51) 

1  .; 

•which  has  the  principal  directions  as  the  directions  of  its  axes  and  the 
square  roots  of  the  absolute  values  of  the  principal  I'adii  of  curvature 
as  the  magnitudes  of  its  axes.  The  conic  may  be  regarded  as  similar  to 
the  conic  in  which  a  plane  inhnitely  near  the  tangent  plane  cuts  the 
sui'face  Avlien  infinitesimals  of  order  higher  than  the  second  are  neg- 
lected. In  case  the  surface  is  concavo-convex  the  indicatrix  is  a  hyper- 
bola and  should  be  considered  as  either  or  both  of  tlie  two  conjugate 
hyperl)olas  that  would  arise  from  giving  ::  jiositive  or  negative  values 
in  (51).  The  point  on  the  sui'face  is  called  elli})tic,  hyperbolic,  or 
parabolic  according  as  the  indicatrix  is  an  ellipse,  a  hyperbola,  or  a  pair 
of  lines,  as  happens  Avlien  one  of  the  principal  curvatures  vanishes. 
These  classes  of  })oints  ctn-respond  to  the  distinctions  definite,  indefinite, 
and  singular  applied  to  the  quadratic  form  rlr  -\-  '2s//k  -f  f/c'^. 

Two  further  results  are  noteworthy.  Any  curve  drawn  on  the  siirface 
differs  from  the  section  of  its  osculating  plane  with  the  surface  l)y 
infinitesimals  of  higher  order  than  the  second.  For  as  the  osculating 
})lane  passes  through  three  consecutive  points  of  the  curve,  its  inter- 
section with  the  surface  ]iasses  through  the  same  three  consecutive 
})oints  and  the  two  curves  have  contact  of  the  second  order.  It  follows 
that  the  radius  of  curvature  of  any  curve  on  the  surface  is  identical 
with  tliat  of  the  curve  in  which  its  osculating  plane  cuts  the  surface. 
The  other  result  is  Meusfilcrs  Tlicorem  :  The  radius  of  curvature  of  an 
obli(pie  section  of  the  surface  at  any  point  is  the  projection  upon  the 
})lane  of  that  section  of  the  radius  of  curvature  of  the  normal  section 
which  })asses  through  the  same  tangent  line.  In  other  words,  if  the 
radius  of  curvature  of  a  normal  section  is  kiu)wn,  that  of  the  oblique 
sections  through  the  same  tangent  line  may  l)e  obtained  1)V  nudti])lying 
it  by  the  cosine  of  the  angle  between  the  plane  normal  to  the  surface 
and  the  plane  of  the  ol)li(]ue  section. 


146  DIFFEEEXTIAL   CALCULUS 

The  proof  of  Meu.snier",s  Theorem  may  be  given  by  reference  to  (48).  Let  the 
r-axis  in  the  tangent  plane  be  talten  along  the  intersection  with  tlie  oblique  plane. 
Neglect  infinitesimals  of  higher  order  than  the  .second.    Then 

y  =  <j,{x)=l  ax",         z-\  (>vf2  +  2  aiy  +  ty")  =  \  rx"  (48') 

will  be  the  equations  of  the  curve.  The  plane  of  the  section  is  az  —  ry  =  0,  as  may 
be  seen  by  inspection.  The  radius  of  curvature  of  the  curve  in  this  plane  may  be 
found  at  once.  For  if  u  denote  distance  in  the  plane  and  perpendicular  to  the 
/-axis  and  if  v  be  the  angle  between  the  normal  plane  and  the  oblique  plane 
az  —  ry  =  0, 

u  =  z  sec  V  —  y  esc  v  =  \r  sec  v  ■  x-  =  \  a  esc  v  •  x-. 

The  form  u  =  \  rsec  v-  x-  gives  the  curvature  as  csec  v.  But  the  curvature  in  the 
normal  section  is  /•  by  (48').  As  the  curvature  in  the  oblique  section  is  sec  v  times 
that  in  the  normal  section,  the  radius  of  curvature  in  the  oblique  section  is  cos  v 
times  that  of  the  normal  section.    Meusnier's  Theorem  is  thus  proved. 

69.  These  investigations  with  a  special  choice  of  axes  give  geometric  proper- 
ties of  the  surface,  but  do  not  express  those  properties  in  a  convenient  analytic 
form  ;  for  if  a  surface  z  =f{x.  y)  is  given,  the  transformation  to  the  .special  axes 
is  difficult.  The  idea  of  the  indicatrix  or  its  similar  conic  as  the  section  of  the 
.surface  by  a  plane  near  the  tangent  plane  and  parallel  to  it  will,  however,  deter- 
mine the  general  conditions  readily.    If  in  the  expansion 

Az-  dz=  I  {rh-  +  2  shk  +  tk-)  =  const.  (52) 

the  quadratic  terms  be  set  ec^ual  to  a  constant,  the  conic  obtained  is  the  projection 
of  the  indicatrix  on  the  x?/-plane.  or  if  (52)  be  regarded  as  a  cylinder  upon  the 
xy-plane,  the  indicatrix  (or  similar  conic)  is  the  intersection  of  the  cylinder  with 
the  tangent  i^lane.  As  the  character  of  the  conic  is  unchanged  Ity  the  projection, 
the  j)oint  on  the  surface  h  elliptic  if  .s-  <  rt.  Ityperbolic  if  .s-  >  rt,  and  parabolic  if 
s-  =  rt.  Moreover  if  the  indicatrix  is  hyperlxilic.  its  asyniptntes  must  project  into  the 
asymptotes  of  the  conic  (52),  and  hence  if  dx  and  d/j  replace  h  and  k,  the  equation 

rdx-  +  2  siJxd)/  +  tdy-  =  0  (58) 

may  be  regarded  as  the  differential  (yuation  of  the  projection  of  the  asymptotic  lines 
on  the  xy-plane.  If  r.  s,  t  be  expressed  as  functions /^^,/^^,/,^,',  of  (j-,  y)  and  (53)  be 
factored,  the  integration  of  tlie  two  ecjuations  ^[{x.  y)dx  +  X{x.  y)dy  thus  found 
will  give  the  finite  equations  of  the  projections  of  the  asynqitotic  lines  and,  taken 
with  the  equation  z  =f{x.  y).  will  give  the  curves  on  the  surface. 

To  find  tlie  lines  of  curvature  is  not  quite  .so  simple  :  for  it  is  necessary  to  deter- 
mine the  directions  which  are  tlie  projections  of  the  axes  of  the  indicatrix.  and 
these  are  not  the  axes  of  the  projected  conic.  Any  radius  of  the  indicatrix  may 
be  regarded  as  the  intersection  of  the  tangent  plane  and  a  plane  perpendicular  to 
the  xy-plane  through  the  raditis  of  the  projected  conic.    Hence 

z-  z,,  =  p  (x  -  x.j)  +  q  {y  -  ?/y),  {x  -  ,/:,;>  k  =  (y  -  y^)  h 

are  the  two  planes  which  intersect  in  the  radius  tliat  projects  along  the  direction 
determined  by  A,  k.    The  direction  cosines 

h:k^vJ^,k ^^_^^      ^^^^^^  ^,^^ 

a7(-  +  /.•■-  +  {ph  -\-  qk)- 


PARTIAL  DIFFEREXTIATIOX  ;   BIPLICIT  147 

are  therefore  those  of  the  radius  in  the  indicatrix  and  of  its  projection  and  tliey 
determine  the  cosine  of  the  anj^le  (f>  between  the  radius  and  its  projection.  The 
square  of  the  radius  in  (52)  is 

/;-  +  A:-,     and     {Ifi  +  k-)sec-4>  =  h-  +  k-  +  {ph  +  qk)'- 

is  tlierefore  tlie  square  of  tlie  corresponding  radius  in  the  indicatrix.  To  deter- 
mine the  axes  of  the  indicatrix,  this  radius  is  to  be  made  a  maxinuuu  or  mininuini 
subject  to  (52).    With  a  nudtiplier  X, 

h  +  ph  +  qk  +  X  {rh  +  .si-)  =  0,         A:  +  pjh  +  qk  +  X  {sli  +  tk)  =  0 

are  tlie  conditions  required,  and  the  elimination  of  X  gives 

ifi  [,s  (1  +  pi)  _  p,/,]  +  ki:  It  (1  +  ;/-)  -'•(!  +  <r)]  -  /^-  [t  (1  +  q-)  -  pqi]  =  0 

as  the  equation  that  determines  the  projection  of  the  axes.    Or 

(1  +  p-)  dx  +  pqdy      pqdx  +  (1  +  q")  dy 

= (oo) 

rdx  +  »dy  sdx  +  tdy 

is  the  differential  equation  of  the  projected  lines  of  curvature. 

In  addition  to  the  a.symptotic  lines  and  lines  of  curvature  the  geodesic  or  shortest 
lines  on  the  surface  are  important.  'I'liese.  however,  are  better  left  for  the  methods 
of  the  calculus  of  variations  (§  15!»).  The  attention  may  tlierefore  be  turned  to 
finding  the  value  of  the  radius  of  curvature  in  any  normal  .section  of  the  .surface. 

A  reference  to  (48)  and  (40)  shows  that  the  curvature  is 

I  _2z  _  rh-  +  2  shk  +  tk-  _  rh-  +  2  shk  +  tk'^ 
R~^~  y^  ^  h-  +  k^ 

in  the  special  ca.se.  But  in  the  general  case  the  normal  distance  to  the  .surface  is 
(Az  —  dz)  cos  7,  with  .sec  7  =  Vl  +  p-  +  q-,  instead  of  the  2  z  of  the  special  case,  and 
tlie  radius  p-  of  the  special  case  becomes  p-.sec'-<?l)  =  h-  +  A:'-  +  {ph  +  qk)-  in  the 
tangent  plane.    Hence 

1  _      2  {Xz  —  '72)  cos  7      _  rl-  +  2  shn  +  tm- 
Tl~~h^+k-  +  {ph  +  qkf~~     Vl  +  p-^  +  1? 

where  the  direction  cosines  I.  in,  of  a  radius  in  the  tangent  iilane  have  been  intro- 
duced from  (54),  is  the  general  expression  fur  the  curvature  of  a  normal  section. 
The  form 

1  rh^  +  2M  +  tk^  1  ..^.,.^ 

—  = —  ,  (ob ) 

R      h-  +  k-  +  {ph  +  qk)-  VI  -(-  pi  +  qi 

where  the  tlirection  /(,  k  id  the  projected  radius  remains,  is  frequently  more  con- 
venient than  (50)  wliicli  contains  tlie  direction  cosines  /.  in  of  the  oriuina!  direction 
in  the  tangent  plane.    Meusnier's  Theorem  may  now  be  written  in  the  form 

cr>s  V       rl-  4-  '2  shn  -f-  tm- 

i'  \^1  +  p-  +  q- 

where  u  is  the  angle  between  an  oblique  .section  and  the  tangent  plane  and  where 
I.  m  are  the  direction  cosines  of  the  intersection  of  the  x'lanes. 

The  w(jrk  here  given  has  depended  for  its  relative  simplicity  of  statement  upon 
the  assumption  of  tlie  .Mirface  (40)  in  solved  form.  It  is  merely  a  prolilem  in 
implicit  partial  differentiation  to  pa.-<s  from  p.  q.  r.  .s.  t  to  their  equivalents  in  terms 
of  F^,  F'    F^  or  the  derivatives  of  0,  ^,  x  by  a,  /3. 


us  DIFFEKEXTIAL   CALCULUS 

EXERCISES 

1       r  +  i      r  —  t 

1.  In  (49)  show  —  = 1 cos  2^  +  5-  sin  2  0  and  find  the  directions  of 

li  2  2 

inaxinuini  and  niininuun  R.    If  U^  and  i?.,  are  the  rnaxinuuu  and  niininiuni  values 
of  li,  show 

11  .  1       1      1  ,         o 

1 =z  r  +  t     and =  H  —  s^. 

Half  of  the  sum  of  the  curvatures  is  called  the  mcdn  riD-vciturc ;  the  product  of  the 
curvatures  is  called  the  total  cunuture. 

2.  Find  the  mean  curvature,  tlie  total  curvature,  and  therefrdni  (hy  construct- 
ing anil  sdlvinn'  a  (juadratic  equation)  tlie  principal  radii  of  curvature  at  the  origin  : 

(a)  z  =  ./■(/.  (,3)  z  =  X-  +  j-t/  +  y-.         (y)  z  =  x  {x  +  y). 

3.  In  tlie  surfaces  (a)  z  =  xy  and  (/3)  z  —  2x-  +  y~  find  at  (0,  0)  tlie  radius  of 
citrvature  in  the  sections  made  by  the  planes 

{a)  x  +  y  =  0.  (/3)  X  +  y  +  z  =  0.  (7)  .r  +  ?/  +  2  z  =  0. 

{5)x-2y  =  0.         (e)x- 2^  +  2  =  0,  (,0  .r  +  2  y  +  U  =  0. 

The  obliijue  sections  are  to  be  treated  by  applying  Meiisnier's  Tlieoreni. 

4.  Find  the  asymptotic  directions  at  (0.  0)  in  Exs.  2  and  ?,. 

5.  SIkiw  that  a  developable  Nurf((ce  is  everyvhere  parahoUe.  that  is.  that  rt  —  s-  =  0 
at  every  point  ;  and  conversely.  'J"o  do  this  considfr  the  surface  as  the  envelope  of 
its  tangent  plane  z  -  p^/  -  7,,^  =  Zy  -  p^^x^^  -  (j„y^,  where  p^,,  q^^,  .r^.  y^.  z^  are  func- 
tions of  a  single  parameter  lx.    Hence  show 

jI'^)  =  0  =  (,., -  .^,.   »,„■   ,,(t'.ii^'i.'.. -'.M  =  ,„(,=  _  ,,)„. 

Vir  t'o/  \  •' [)•  If  I)  I 

The  first  result  proves  the  statement  ;  the  second,  its  converse. 

6.  Find  the  difft'reiitial  equations  of  tlu'  asymptotic  lines  and  lines  of  curvature 
on  these  surfaces  : 

(a)  z  =  xy,         ip)  z  =  tan-i(;///),         (7)  z-  +  y-  =  cosh.f,         (5)  xyz  =  1. 

7.  Siiow  that  the  mean  curvature  and  total  curvature  ari' 
1/]        J\_r{l  +  q-^)  +  t{l+p^)-2pqs  1      _         rt-s- 


8.  Find  the  j)rincipal  radii  of  cur\ature  at  (1.  V)  in  l".x.  (>. 

9.  An  umbilic  is  a  point  of  a  surface  at.  whicii  tlie  i)rincipal  radii  of  curvature 
(and  hence  all  radii  of  curvatui-e  for  normal  .sections)  are  equal.    Show  that  the 

conditions  are =  -    = for  an  umV)ilic,  and  determiiR'  the  umbilics  of 

1  +  /'-       /"/       1  +  q- 
tlie  ellijjsoid  with  semiaxes  a,  h,  c. 


CHAPTER   VI 

COMPLEX  NUMBERS  AND  VECTORS 

70.  Operators  and  operations.  If  an  entity  u.  is  changed  into  an 
entity  r  by  sdhk^  law.  tlu'  clianyv  may  be  regarded  as  an  ojifrdtina  per- 
formed upon  a,  the  (ipcninil.,  tcj  convert  it  into  r;  and  if /'he  introdneed 
as  tlie  syndtol  of  the  (j])eration,  the  resnlt  niay  Ije  written  as  /•  =./'". 
For  Ijrevity  the  syjidx)!  /'  is  often  ealled  an  oprmfor.  A'arions  sorts 
of  operand,  o}»erat(jr,  and  i-esult  are  familiar.  Thus  if  n  is  a  })(jsitive 
number  n,  the  a})plieation  of  the  operator  ^  gives  the  square  root:  if  u 
represents  a  range  of  values  of  a  vaiiable  .'■,  the  expressi(jn /'(./■")  or  f.r 
denotes  a  function  of  .r ;  if  v  lie  a  function  of  ./•,  the  operation  of  diC- 
fereiitiation  may  be  syndioli/.ed  by  1)  and  the  result  ])ii  is  the  deriva- 
tive ;   the  symbol  of  definite  integration    |     (*)'/*  converts  a  function 

i'(.'-)  into  a  numbei';  and  so  on  in  great  variety. 

The  reason  for  making  a  short  study  of  operators  is  that  a  consider- 
able num!)er  of  the  concepts  and  i-ules  of  arithmetic  and  algebra  mav 
be  so  defined  for  operators  themselves  as  to  lead  to  a.  ciilcuhin  of  oprrii- 
tiiins  which  is  of  fre(juent  use  in  matliematics  ;  the  single  a}»plication  to 
the  integration  tif  certain  diil'erential  e(juations  (§!>."»)  is  in  itself  highly 
valuable.    The  fundamental  conce})t  is  that  of  'a  prail m-f :  If  i'  ''•">■  oj^icr- 

(tfi'il  upnil  h(j  f  fi)  /jii-!'  fil  =  C  rind  if  C  (S  iijicrdtt'd  lljiojl  In/  (J  tit  Ij'li'i'  [J  i'  =  ir, 
sn  thilt 

J"  —  '',       [/''  = ://"  =  "',       at"  =  "',  (1) 

f/ir/i  till'  opi'l'iitinii  'uLiliniti'il  US  ijf  ir]iii-Ji  cnnri'iis  II  (I'u'pi-tlii  into  ir  is 
riilli'iJ  till'  proihirt  of  f  III/  [/ .  If  the  functi<jnal  syml)ols  sin  and  log  be 
regarded  as  operators,  llie  symbol  log  sin  ccndd  be  regarded  as  the 
product.  The  transi'ormations  of  turning  the  ,i-ij-\A-a\w  over  on  the 
./•-axis,  so  tliat  .'•'  =  ,/•,  //'  =  —  _//,  and  over  the  y-axis,  so  that  ./•'  =  —  ,/■, 
//'  =  _y,  may  be  regarded  as  operations:  the  cond)ination  of  these  o})era- 
tious  gives  tlu^  ti-ansformation  ,/■'  =  —  ./•,  //'  =  —  y,  which  is  erpiivalent 
to  rotating  the  jilaiie  through  180°  about  the  origin. 

The  products  of  ai-ithmetic  and  algebra  satisfy  the  com nintntiri'  Imr 
i/f  =  fi/^  tliat  is.  tlie  products  of  y  In"  /'and  of _f  b\'  y  are  etpial.  This 
is  not  true  of  operators  in  geutu'al,  as  n)av  be  seen  from  the  fact  that 

14',> 


150  DIFFEREXTIAL  CALCULUS 

log  sin  X  and  sin  log  x  are  different.  Whenever  the  order  of  the  factors 
is  inmiaterial,  as  in  the  case  of  the  transformations  just  considered,  the 
ojjerators  are  said  to  he  romvivtatlri^.  Another  hnv  of  arithmetic  and 
algebra  is  that  when  there  are  three  or  more  fa(-t<jrs  in  a  })roduct,  tlie 
factors  may  be  grouped  at  pleasure  without  altering  the  result,  that  is, 

This  is  known  as  the  ossorlatlrc.  hm-  and  operators  Avhich  obey  it  are 
(•alhnl  (issocitifirc.  Only  associative  operators  are  considered  in  the 
work  here  given. 

For  the  repetition  of  an  operator  several  tinu^s 

ff = A    fry = A    AT  = ./'"' + ",  (3) 

the  usual  notation  of  powers  is  used.  TI/c  Imr  nf  in'fii-rs  rlcarl//  holds; 
for  f"'  +  "  means  that  /'  is  applied  //>  +  /;  times  successively,  whereas 
/"*/'"  means  that  it  is  applied  ?i  times  and  then  ///  times  more.  Xot 
ap})lying  the  operator /'at  all  would  naturally  be  denoted  by/"",  so  that 
f^i/  =  u  and  the  operator/^  would  ))e  ecpiivalent  to  multiplication  l)y  1; 
the  notation  /^  =  1  is  adopted. 

If  for  a  given  operation  f  there  can  be  found  an  opei'ation  (/  such 
that  the  product  f(j  =  f'  =  l  is  equivalent  to  no  ojieration,  then  g  is 
called  the  Inverse  of /'and  notations  such  as 

f'j  =  ^,    (i=A'  =  Y   A-'=fy-^  (4) 

are  regularl}'  borrowed  from  arithmetic  and  algebra.  Tlius  the  inverse 
of  the  square  is  the  square  root,  the  in\-erse  ol'  sin  is  sin~',  the  inverse 

of  the  logarithm  is  the  exponential,  tlie  inverse  of  />  is    /.     Some  ojter- 

ations  have  no  inverse;  multiplication  liy  0  is  a  case,  and  so  is  the 
s(|uare  when  applied  to  a  negative  number  if  only  real  numliers  are 
considered.  Other  operations  have  more  than  one  inverse;  integra- 
tion, the  inverse  of  D,  involves  an  arbitrary  additive  constant,  and  the 
invei'se  sine  is  a  multiple  valued  function.  It  is  tlierefore  not  always 
true  tliat/'~\f  =  1,  but  it  is  customary  t(^  mean  by  /'^^  that  ])ai'ticular 
inverse  of /' for  which  /'"■'/' =  //'~"^  =  L  Higher  negative  ])0\vei's  are 
dehned  by  the  e(|uation  /"~"  =  ('/■"^)",  and  it  readily  follows  that 
/■'/'-"  =  1,  us  may  l)e  seen  by  the  exanqjle 

77//'    /ilir    f>j'  i/l(/irrs  /'"/"=/'"'  "    ii/sn    hnlds    fnr    Drijilt i rr    iu'llrrs.    eXce]it 

in  SO  far  as /'"'/' nia\"  not  be  e(pial  to  1  and  may  l)e  reipiiii'd  in  the 
reduction  of  /'"/'"  to  /-'"'  + ". 


COMPLEX  NUMBERS  AXD   VECTORS  151 

If  u,  V,  and  u  +  v  are  operands  for  the  operator  /  and  if 

A" +  0= /"+/';  (5) 

so  that  the  operator  applied  to  the  sum  gives  tlie  same  result  as  the 
sum  of  the  results  of  operating  on  each  operand,  then  the  operator 
f  is  called  linear  or  disfrihutice.  If  /  denotes  a  function  such  that 
y(./-  -)-  _//)  —f{.r)  -j-f(t^)^  it  has  been  seen  (Ex.  9,  p.  45)  that  /  must  be 
equivalent  to  multiplication  by  a  constant  and  fx  =  Cx.  For  a  less 
specialized  interpretation  tins  is  not  so ;   for 

D(ii  -f-  r)  =  Di(  4-  T)r      and     (   (tf  +  r)  =    I  "  +    /  '' 

are  two  of  the  fundamental  formulas  of  calculus  and  sliow  operators 
which  are  distributive  and  not  equivalent  to  multiplication  l)y  a  constant. 
Nevertheless  it  does  follow  l>y  the  same  reasoning  as  used  before  (Ex.  9, 
p.  45),  t\iH,t  full  =  72f('  if /is  distributive  and  if  7i  is  a  rational  number. 
Some  operators  have  also  the  property  of  addition.  Suppose  tliat  ii 
is  an  operand  and/',  r/  are  operators  such  that  fi(  and  f/it  are  things  tliat 
may  be  added  together  as /V  +  ;/i/,  tlien  the  sum  of  tlie  operators, /+  y, 
is  defined  by  the  equation  (/-'-[-,'/)"  =/V  +  ,'/"•  If  furthermore  the 
operators  /',  (/,  h  are  distributivf,  then 

//  (f  +  y)  =  If  +  Ay     and     (/'  +  y )  h  =  f/i  +  r/Ji ,  (G) 

and  the  multiplication  of  the  0})erators  becomes  itself  distributive.  To 
jjrove  tliis  fact,  it  is  merely  necessary  to  consider  that 

/'  [(,/+  r/) "]  =  ^'  (/"  +  !/")  =  ¥'"  +  /','/« 
and  (f  +  'J)  (Ji  I')  =  fli "  +  'J^'  "• 

Operators  irlta-lt  are  assoelafire^  cnmriuitatlre,  disfribntive,  arid  irlia'h 
admit  addition  ma  ij  he  treated  (ihjct/ralraJJ  ij^  in  so  fir  as  pohpioinlojs  are 
einxferned .  hi/  tlie  ordlna rij  ahjorlsins  of  ahji'hra  ;  for  it  is  by  means 
of  the  associative,  commutative,  and  distrilnitive  laws,  and  the  law  of 
indices  that  oi'dinary  algel»raie  })olynomials  are  rearranged,  multi})lied 
out,  and  factored.  N(jw  the  operations  of  multi])lication  l)y  constants 
and  (jf  dift'erentiatiou  or  ])artial  ditt'erentiation  as  a])plied  to  a  function 
of  one  or  more  variables  ,/•,  //,  z,  ■  •  ■  do  satisfy  tliese  laws.    For  instance 

r(lJ(()  =  D(i-)l').     ]>J»„ii  =  ^>J',";     ( l'.r  +  l>,i)I>:"  =  I  >,!>■"  +  l'„^'z>'-     (~) 

Hence,  for  example,  if  //  be  a  function  of  ./■,  tlie  expression 

J>"i/  +  "J>"-^!J  -\ h  "u^J>if  +  ".,!/, 

where  the  coefficients  a  are  constants,  mav  be  written  as 

(^Tr  +  ajr^-^^^-...  +  n_^_j>Jr",oij  (8) 


152  DIFFEKEXTIAL   CALCULUS 

and  may  then  be  factored  into  the  form 

[i^D-  a  J  {^JJ  -  aj  ...{J>~  a,^  -,){D-  «,  j]  y,  (8') 

where  a^,  (X,„  •  ■  •,  «„  are  the  I'uots  of  the  algel)raic  polynomial 

EXERCISES 

1.  Show  that  (fgh)-'^  =  Ji-'^y-'^f-^,  tliat  is,  that  the  reciprocal  of  a  profluct  of 
operations  is  tlie  product  of  tlie  reciprocals  in  inverse  order. 

2.  ISy  (It'linitidn  the  operator  y/'y-i  is  called  the  transform  offhyg.  SIkiw 
that  (a)  the  transform  of  a  product  is  the  product  of  the  transforms  of  the  factors 
taken  in  the  same  order,  and  (p)  the  transform  of  the  inverse  is  the  inverse  of  the 
transform. 

3.  If  .s  :7f:  1  but  .s-  =  1,  the  operator  ,s  is  by  definition  said  to  be  involutory .  Show 
that  {a)  an  involutory  operator  is  equal  to  its  own  inverse;  and  Cduversely  (/3)  if 
an  operator  and  its  inverse  are  equal,  the  f)perator  is  involutory  ;  and  (7)  if  the 
product  of  two  invdlutory  operators  is  connmitative,  the  jn-oduct  is  it.self  involu- 
tory ;  and  conversely  (5)  if  the  product  of  twu  involutory  operatcjrs  is  involutory, 
the  operators  are  conmuitative. 

4.  If/' and  ij  are  both  distributive,  so  are  the  products/f/  and  gf. 

5.  If /is  distributive  and  n  rational,  sliow/x;/  =  nfu. 

6.  Expand  the  followiuic  operators  first  by  ordinary  formal  multiplication  and 
second  by  applying  the  operators  successively  as  indicated,  and  show  the  results 
are  identical  by  translating-  both  into  fanuliar  forms. 

(a)   (/>-l)(/>_2)y.     Ans.  '^'^^  _  8 '|^  +  2  y. 

(/5)  {I)-}}1){JJ  +  1 )  //,  (7)  U  (1)  -  2)  (7J  +  1)  (Z»  +  3)  y. 

7.  Show  that  (I)—  ii)\  <-"■''  i  (.—  "'A'lU  =  A',  where  .V  is  a  function  of  x,  and 
heucf  infer  that  t"''  /  (-'"(:•;)'/./•  is  the  inverse  of  the  operator  {I)  —  '()(*). 

8.  Show  that  ])[i"'  IJ)  =  i"'{])  -\-  <i)  IJ  and  hence  i;cneralize  to  sliow  that  if 
L'(l))  denote  any  polynomial  in  1)  with  constant  coeliicii'Uts.  tlieii 

7^(7;)  •  ("■'[/  =  (".'■P(7V+  u)y. 

Apply  this  to  the  followinu-  and  check  the  results. 

(a)   (/>-  -:!/;+  2),-  ■>/  ^  e^-(Ifi  +  ]))y  =  e--i'p^  +  '^) , 

(,:!){ I  J-  -  ■■]  J)  --  -Ix  'y.  (7)    ( Ir  -  ;•]  7^  +  2 )  c'y. 

9.  ]f  //  is  a  function  of  x  and  x  =  ('  slio\v  that 

]>,y  =  e-'J),y,  jyj.y  =  v--'l),{I),-  })y.  ■■■.  ])';y  =  (-''' I),{I),-  \)---{I),-p  +  1)//. 

10.  Is  the  expression  i/il),.  +  IrJ),,)".  which  occurs  in  Taylnr's  Koiinulii  (^  .')4). 
the  ),tli  power  of  the  (i])erator  /il),  +  1:1),,  or  is  it  merely  a  conventional  .-^yndiol  V 
'I'Jic  .-ame  que.-tion  relatixe  to  (,;■/>,.  +  yl),i]'-  (iccurrin::'  in  I-;uler">  l-'nrnuihi  ( ,'   '>'■))  '.' 


COMPLEX  NUMBERS   AX])   VECTORS  15:3 

71.  Complex  numbers.  In  the  formal  solution  of  the  equation 
't.r-  +  //./•  4"  '■  =  0,  where  Ir  <  4  c/c,  uumljers  of  the  form  ///  +  a  V—  1, 
where  in  and  n  are  real,  arise.  Such  numlxu's  are  called  cuniplcf  or 
imar/lnanj  ;  the  part  in  is  called  the  real  pit rt  and  n\  —  1  Ww.  i)nre 
InuKjmanj  piirt  of  the  number.  It  is  customary  to  write  v—  1  =  l  and 
to  treat  i  as  a  literal  quantity  subject  to  the  relation  r  =  —1.  The  defini- 
tions for  the   cquardij,  addition,  and  inultijilicdtinn  of  conq)lex  num- 

l)ers  are  , .  ,.     -n       t       i     -r 

a  -\-  Oi  =  r  -\-  a  I.     it  and  only  it     a  =  r,  h  =  a, 

[./,  +  /./]  +  [..  +  di-]  =  (ii  +  r)  +  (h  +  d)  i.  ,    (9) 

[a  -f  /'>/]  [r  4-  r/i]  =  (,ir  —  />d)  +  Qid  +  hr)  /. 

It  readily  follows  that  f/ie  rn/nmufafin',  (fssoeiatirr,  and  distributire 
/airs  /add  in  the  domain  of  i-omplcx  nHndicrs,  namely, 

«  +  ^  =  /3  +  «,  (,j;  +  ^)  -f.  y  =  «  +  (/3  +  ^)^ 

al3  =  l3'C,  (a^)y  =  a{/3y).  (10) 

aiJB  -\-  y)  =  a (3  +  ay,  (ft  -{-  /Sjy  =  ay  +  I3y. 

where  Greek  letters  have  Ijeen  used  to  denote  com])lex  nuniliers. 
JJirision  is  accomplished  Ijy  the  method  of  rationalization. 

a  +  ///  _a  ^  hi  r  —  di  _  (<if  +  hd )  +  (/>r  —  <id)  i 

e  -f-  di         r  +  di  r  —  di  <■'-  +  </-  ^ 

This  is  always  possible  except  Avlien  r'^  +  ^/'- =  0,  that  is,  wlien  Ixith  r 
and  '/  are  0.  A  conijilex  miiuber  is  defined  as  0  wlien  and  only  when 
its  real  and  i)ure  imaginary  parts  are  l)otli  zero,  ^\'itll  this  delinition  0 
has  the  ordinary  jiroperties  that  a  +  0  =  a  and  «0  =  0  and  that  a/O  is 
im})Ossible.  Furthermore  if  n  jimdiicf  a/3  cnnislifs,  ciflwr  a  or  /3  canisln-s. 
For  suppose 

\_a  +  />/]  [r  +  </;]  =  (<,r  -  hd)  +  (ad  +  l,r)  i  =  0. 

Then  ar  -  hd  =  0     and     ad -\~ />r  =  0,  (12) 

from  which  it  follows  tliat  eitlier  a  =  //  =  0  or  c  —.  d  =  0.  From  the 
fact  that  a  product  cannot  vanish  unless  one  of  its  factors  vanishes 
follow  the  ordinary  laws  of  cancellation.  In  l)rief,  "//  t/w  rln/u^ntari/ 
lairs  of  rriil  alijrhr<i  ludd  also  for  tlw  aljjchra  of  i-inn ph'X  naiiihrrs. 

By  assuming  a  set  of  Cartesian  coordinates  in  the  .'■//q)lane  and  asso- 
ciating the  number  a  -j-  hi  to  the  point  (a,  Ji),  a  grapliical  rcprrsmfafion 
is  obtained  which  is  the  counterpart  of  the  number  scale  for  real  num- 
bers. The  point  (a.  //^  alone  or  the  directed  line  from  the  (jrigin  to  tlie 
])oint  (a,  h)  may  be  considered  as  representing  tlie  number  '/  +  ///. 
If  ()/'  and  (>(}  are  two  direeted  lines  repi-esenting  tlie  two  numbers 
a  4-  /'i  and  '•  +  di,  a  referenee  to  thi-  iigure  shows  that  the  line  wliich 


154 


DIFFERENTIAL   CALCULUS 


(a+c,b  +  cl) 


represents  tlie  sum  of  the  iiumhers  is  0/!,  the  diagonal  of  the  paralleh> 
graiii  of  ^vhieh  OP  and  0(2  are  sides.  Thus  fJiP  (jcomcfrlr  hnr  fnr  mhlbuj 
coiiiph'X  numhura  is  f/ie  saiiif  "s  flw  hnr  fur  cniii^xrnndln'/  farci-s  (md  is 
hnoirn  as  tlie  pdrdlh'lorjrdiii  latr.  A  segment  AB  of  a  line  possesses 
magnitude,  the  length  All.  and  direetion,  the 
direction  of  the  line  AFi  from  .1  to  B.  A 
quant'dij  vlilch  Ikis  rniif/n'ifn<h'  find  dirrcfhni  is 
n/lli-d  II  rrrtiir  :  mid  tlir  piiriilh'lixjnnn.  hnr  is 
cdUed  tJw  hnr  of  rertur  addition.  ( 'oiiiph'.r  nmn- 
bcrs  may  therefore  Ix'  regarded  as  rrctcjrs. 

From  the  figure  it  also  aj)p('ars  that  OQ  and  PPi.  have  the  same  mag- 
nitude and  direction,  so  tliat  as  vcctoi-s  tlicy  are  e(|Ual  although  they 
start  from  different  points.  As  OP  +  J'J!  will  be  regardc^l  as  e([ual  to 
OJ^  +  O'l,  the  definition  of  addition  may  l^e  given  as  the  triangle  law 
instead  of  as  the  parallelogram  law ;  namely,  from  the  terminal  end  J' 
of  the  first  vector  lay  off  the  second  vector  PR  and  close  the  triangle 
hy  joining  the  initial  end  O  of  the  hrst  vector  to  tlie  ternunal  end  R  of 
the  second.  The  idisoliftr  nilni'  of  a  complex  numher  <i  -f  hi  is  the 
magnitude  of  its  vector  Ol'  and  is  ecpial  to  V'r  +  /r,  the  S(|uai-e  root  of 
the  Sinn  of  the  squares  of  its  rt^al  part  and  of  tli(^  coethcient  of  its  })Ui-e 
imaginary  part.  The  absolute  value  is  denoted  by  \(i  +  /'/as  in  the  case 
of  reals.  If  a  and  /3  are  two  complex  niuabers,  tlie  rulc^  't'  +  j8^  ''t  +  /3! 
is  a  consequence  of  the  fai't  that  one  side  of  a  triangle  is  less  than  tlie 
smu  of  the  other  two.  If  tlie  absolute  value  is  given  and  tht^  initial  end 
of  tlie  vector  is  fixed,  tlie  ttM-minal  end  is  thereliy  constrained  to  lie 
u])()n  a  circle  conciMitric  with  tlie  initial  end. 

72.  \N'hen  the  com])lex  numbers  are  laid  otf  from  the  oi'igin,  ])olar 
coordinates  may  be  used  in  ])lace  oi  Cartesian.    Then 

r  =  V"-  +  //-, 
and 


=  tan-'A^'^/*,        II  =  /•  CDS  cf),        />  =  r  sin 
11  -\-  II,  =  /•((•OS  (^  +  /  sin  <^). 


(13) 


The  absolute  value  /■  is  (jften  called  the  iimd nl us  or  nuiijnifiidv  of  the 
coiii])lex  numbei-;  the  angle  (^  is  called  the  iinijh'  or  a r'/nnifnt  of  the 
number  and  suffers  a  certain  indetermination  in  that  2  mr.  where  ii  is 
a  ]iositi\'e  or  negative  integer,  mav  be  added  to  <^  without  aifecting  the 
number.  This  jiolai-  n-preseiitation  is  ])articularly  usefTil  in  discussing 
jiroducis  and  (juotients.     Foi-  if 


a  =  /'jicos  (^j  +  /sin  cf>^).  (S  =  /•.,(cos  ^.,  +  /sin  </>._,), 

then  a/3  =  /y.,  [eos  (^^  +  ^ ,)  -f  /  sin  ( <p^  +  <^.()], 

*  As  both  cos  •■'  uiiil  .sill',''  arr  known,  tin-  (iuailraiit  nf  tliis  aiigli-  is  (Iftfriniin 


14; 


COMPLEX   Xr.^IBEKS   AXJ)   VECTOKS  155 

as  may  be  seen  by  multiplication  aeeordin,t,f  to  the  rule.  Henc-e  the 
1)1(1  (jn'ifudo  of  (I  in'iKJiicf  is  tlie  pro'/i/rf  of  fl/r  vKiynitmhs  of  flic  factors^ 
and  fJie  awjlc  of  <i  [n'oduct  is  the  sii m  of  tJin  (ingles  of  the  factors ;  the 
general  rule  being  i)rove(l  by  induction. 

The  interpretation  of  m iilt'qdicatlon  Inj  a  coinph'.r  niimhcr  as  an  ojicr- 
iitlon  is  illuminating.  Let  ^  he  the  multiplicand  and  a  the  midtiplier. 
As  the  product  a/3  has  a  magnitude  equal  to  the  product  of  the  magni- 
tudes and  an  angle  equal  to  the  sum  of  the  angles,  the  factor  a  used  as 
a  multiplier  may  be  interpreted  as  effecting  tlie  rotation  of  ^  through 
the  angle  of  a  and  the  stivtching  of  /3  in  the  ratio  [a] :  1.  From  the 
geometric  viewpoint,  thei'efoi'c,  iiniJtlpUcdt'ton  luj  u  complcj'  nidiiJier  is 
(in  opcvdtion  (f  rotdtlon  (hkI  strcti-Ji'oKj  In  tlic  phnic.  In  the  case  of 
«  =  cos  ^  +  /  sin  (^  ■\vith  /'  =  !,  the  o})eration  is  only  of  rotation  and 
hence  the  factor  cos  ^  -|-  I  sin  <^  is  often  called  a  cyclic  factor  or  versor. 
In  particular  the  number  i  =  V—  1  will  effect  a  rotation  through  90° 
when  used  as  a  multi})lier  and  is  known  as  a  quadrantal  versor.  Tlie 
series  of  j)Owers  /,  r  =  —  ],  ?"  =  —  /,  /"*  =  1  give  rotations  through  90°, 
180°,  270°,  360°.  This  fact  is  often  given  as  the  reason  for  laying  off 
pure  imaginary  numbers  hi  along  an  axis  at  right  angles  to  the  axis 
of  reals. 

As  a  particular  product,  the  7ith  })Ower  of  a  complex  nmuber  is 

a"  =  (/(.  +  di)"  =  [/'('cos  4>  -\-  i  sin  ^)]"  =  /•"  (cos  n<f)  +  ''  ^in  fi(t>)  ;     (15) 

and  (cos  cji  -\-  i  sin  cf))"  =  cos  n(f>  +  ''  sin  /icf),  (15') 

which  is  a  special  case,  is  known  as  I/c  Jfoim's  T]/corcin  and  is  of  use 
in  evaluating  tlie  functions  f)f  nc^:  foi'  tlie  binomial  theoi'em  may  l»e 
ap})lied  and  the  real  and  imaginary  parts  of  the  expansion  may  be 
equated  to  cos  »<^  and  sin  nc^.    Hence 

V  (n  —  1) 
cos  nc^  =  cos"<^ — cos"    -<^  snr^ 

H — ; ■ cos"    ^(^siir^  — •••  (16) 

-i    , 

_,        .  n(n  —  l)(n  —  2)  „       .    ., 

sm  n(p  =  11  cos"    'c/>  sin  </> ^-; cos"~"^  sin  c/)  +  ■  •  •. 

o  . 

As  the  ??th  root  Va  of  a  must  be  a  numl>er  which  Avhen  raised  to  the 
?;th  power  gives  (X.  the  ??th  root  may  be  written  as 

\a  =  ■y/i-CcoH  cf}/n  +  i  sin  (f>/n).  (17) 

The  angle  cf),  however,  may  have  any  of  the  set  of  values 

(/).      c^  +  2  7r.      cji+iTT.      ■■•.      -/>  +  2(/;  -  l)7r, 


c^    ,    2  7r 

(h          4  TT 

-H , 

-  +  ■ 

11         n 

71         n 

156  IVIFFEKENTIAL   CALCULUS 

anil  the  ?ith  parts  of  tlicso  give  tlic  n  different  angles 

*  +  ^(!iIzilZ.  (IS) 

1  Fence  there  niay  ])e  found  just  71  different  ?itli  roots  of  any  given  com- 
plex number  (including,  of  course,  the  reals). 

The  roots  of  unity  {Icscrve  iiu'iition.  Tlie  eiiuation  x"  =  1  lias  In  the  real  domain 
one  oi-  two  roots  acc'ordinn'  as  n  is  odd  or  even.  But  if  1  be  regarded  as  a  complex 
nundxT  of  which  tlie  pure  imaginary  part  is  zero,  it  may  l)e  represented  by  a  point 
;'t  a  unit  distance  fi-om  the  origin  ujion  the  axis  of  reals;  the  magnitude  of  1  is  1 
and  the  angle  of  1  is  0,  'Itt,  •  •  •,  2(?i  —  l)7r.  The  nth  roots  of  1  will  therefore  have 
the  magnitude  1  and  one  of  the  angles  0,  2  tt/ji,  •  •  • ,  2  {n  —  1)  ir/n.  'i'he  n  nth  roots 
are  therefore 

27r       ..27r        ,  Air       .   .    iir 

1       (t  =;  cos H 'i'SUi  —  ,   (I- =  COS        +  t^ni — ,   •••, 

H  n  n  n 

2(h-  l)7r       .   .    2(u-l)7r 

(r"-i  —  cos—-         ' 1-  tsm-  -  — --, 

?i  n 

and  may  l)e  evalualcd  with  a  table  of  natural  functions.  Now  x"  —  1  =0  is  factor- 
able as  (,c  —  !)(,/■" -i  +  ,(,■"--  +  •  •  •  +  .c  +  1)  =  0,  and  it  therefore  follows  that  the 
i(th  roots  otlier  than  ]  iinist  ;i]l  satisfy  the  eijuation  formed  by  setting  the  second 
factor  ecjual  to  0.  .\s  it  in  particular  satisfies  this  ecjuation  and  the  other  roots  are 
a'-,  •  •  -,  a"~^  it  follows  that  the  sum  of  the  n  nth  roots  of  luiity  is  zero. 

EXERCISES 

1.  Prove  the  distributive  law  of  multiplication  for  conqilex  munbers. 

2.  By  detinitioii  the  jiair  of  imaginarics  (/  +  hi  and  a  —  hi  are  calU'd  ronjiigntc 
imiuiindrics.  Prove  that  {a)  the  sum  and  the  product,  of  two  conjugate  imaginarics 
are  I'eal  ;  and  conversely  (/3)  if  the  sum  and  the  product  of  two  imaginarics  are  both 
real,  the  imaginaries  are  conjugate. 

3.  Show  that  if  /'(,/■.  //)  is  a  symmeti'ic  iiolyudmial  in  x  and  y  with  n^al  coetti- 
cicnts  so  that  l'(.r.  //)  ---  l'(y.  x).  tlien  if  conjugate  imaginarics  be  substituted  forx 
and  /y,  the  value  of  the  polynomial  will  be  I'eal. 

4.  Show  that  if  a -\- Jti  is  a  rout  of  an  algebraic  e(iuation  V{x)  —  0  with  real 
coeflicicnts,  then  a  —  l)i  is  also  a  root,  of  the  eipiation. 

5.  Carry  out  tlir  indieatcd  ojx'rations  algebraically  and  make  a  graphical  repre- 
sciitalioii  for  every  nunilicr  conrcriieil  and  for  the  answer  : 

(a)  {1  +  iy\  (p)  (l  +  V  '.^  /)  (1  -  0.         (7)  (-i  +  ^■  ^V  (^  +  V^), 

V2  -  i  ■\'-i 

-  ')- 

6.  Plot  and  tind  the  modulus  and  angle  in  the  following  cases: 
(<()   -  2,  iji)   -  2  V^l,  (7)   ^l  +  ii,  (5)    I-  I  n'^. 


<^>  ii  ;• 

1  -  i  -X'l-i 

iv)    ,,    ,    .  .,' 

(1  +  it' 

COMPLEX  NUMBERS   AXD  VECTORS  157 

7.  Sliow  that  iJie  modulus  of  a  quotient  ofttco  numhvrf^  is  the  (juolient  of  tite  moduli 
and  that  the  angle  is  the  anrjle  of  the  numerator  leas  thai  of  the  denominator. 

8.  Carry  out  the  indicated  operations  trigononictrieally  and  plot: 

(a)  Tlie  examples  of  Ex.  5,  (/S)  Vl  +  i  \'l  -  (,  (7)  \/-  2  +  2\^Si, 

(5)  ( VITl  +  Vl^)-,  (e)   VV2  +  V^,  (s-)  \2  +  -^  VS  (, 

(77)  ^10  (cos  200^^+  (sin  200°),  (^)  -v/ITT,  (0  "v^. 

9.  Find  the  ecpiations  of  analytic  j^eonietry  whicli  represent  the  transforma- 
tion ecjuivalent  to  nudtiplication  by  d  =  —  1  -\-  V—  o. 

10.  Show  that  l^  —  a  I  =  /•,  where  z  is  a  varial)le  and  a  a  fixed  complex  mimber. 
is  the  equation  of  the  circle  (,r  —  (/)-'  +  (//  —  b)-  =  /•-. 

11.  Find  cosox  and  cos8.f  in  terms  of  cosj-,  and  sinO  j  and  sinTj  in  terms  (if 
sin  .r. 

12.  Obtain  to  four  decimal  places  the  five  mots  VI. 

13.  If  z  =  J'  +  iy  and  z'  =  x'  -\-  i;/',  show  that  z'  =  (c<is0  —  /sin  (p)z—  a  is  the 
formula  for  shifting  the  axes  tiirougli  the  vector  distance  if  =  a  +  ih  to  the  new 
origin  (a,  h)  and  turning  them  through  the  angle  (p.  Deduce  the  ordinary  ecjua- 
tions  of  transformation. 

14.  Show  that  \z  —  "1=  A-jz  — /3|.  where  I  is  real,  is  the  e(]natiiin  of  a  circle; 
spt'cif}^  the  piisitiiin  of  tlie  circle  carefully.  I'si'  the  theorem  :  The  locus  of  points 
whose  distances  to  two  lixed  points  are  in  a  constant  ratio  is  a  I'ircle  the  diameter 
of  which  is  divided  internally  and  externally  in  the  same  ratio  by  the  tixcd  points. 

15.  The  transformation  z'  =  — ,  where  a.  h.  r,  d  are  conqilex  and  ad—  he  ^  0, 

cz  +  d 
is  called  the  general  linear  transformation  of  z  into  z' .    Show  that 

ra  +  '/' 
\z' —  a'\  =  k\z' —  p'\     becomes     \z  —  a\=k  .-    -  -   ;-,z— /3|. 

I  r;i  +  f/ 1 

Hence  infer  that  tlie  transformation  cari'ies  circles  into  circles,  and  points  wliich 
divide  a  diameter  internally  and  externally  in  the  same  ratio  into  jioints  whicli 
divide  some  diameter  of  tlie  new  circle  similarly,  but  generally  with  a  dit't\'rent  I'atio. 

73.  Functions  of  a  complex  variable.  Lot  ,v  =  ,/•  +  ///  lie  a  coinplcx 
variable  reprcseiitalilo  yooiuotrically  as  a-  varialilo  point  in  the  ;''//-plaiio, 
which  may  he  called  the  rmiipler  jilanc  As  z  (leteniiiiies  the  two  real 
numbers  .-'•  and  //,  any  ftmction  Fi-r,  //)  which  is  the  sum  of  two  single 
valued  real  functions  in  the  form 

F{.r,  ,/■)  =  X  (./•.  v/)  +  IV {.r,  //)  =  Jl  (cos  4)  +  (•  sin  ^I>)  (19) 

will  be  cotupletely  determined  in  value  if  z  is  given.  Such  a  function 
is  called  a  cooijtJe.i'  function  (and  not  a  function  of  tlie  eom})le.x:  vjirialjle, 
for  reasons  that  will  appear  later).  The  magnitude  and  angle  of  the 
function  are  determined  by 

.V      .  ]' 


/,'  =  Va'-  +  Y\         cos  $  =  -7 ,  sin  *  =  -  .  (20) 


158  DIFFERENTIAL  CALCULUS 

The  function  F  is  continuous  Ijy  definition  wlien  and  only  -vvlien  both 
X  and  Y  are  continuous  functions  of  (,r,  y)  ;  7.'  is  then  continuous  in 
(./•,  y)  and  F  can  vanish  only  when  R  =  0 :  the  angle  $  regarded  as  a 
function  of  (.7-,  ?/)  is  also  continuous  and  determinate  (except  for  the 
additive  2  inr)  unless  R  =  0,  in  Avhich  case  X  and  Y  also  vanish  and  the 
expression  for  $  involves  an  indeterminate  form  in  two  variables  and 
is  generally  neither  determinate  nor  continuous  (§  44). 

If  the  derivative  of  F  with  respect  to  z  were  sought  for  the  A'alue 
z  =  a  +  >^>,  the  procedure  Avould  be  entirely  analogous  to  that  in  tlie 
case  of  a  real  function  of  a  real  variable.  The  increment  A,t'  =  A,/'  +  /Ay 
would  be  assumed  for  z  and  AF  would  be  computed  and  the  quotient 
AF/Az  would  be  formed.    Thus  by  the  Theorem  of  the  ]Mean  (§  4(3), 

AF ^  AA'  +  ;a )' ^  (x:  +  / );;.) \.,-  +  (x;  +  ; )-;) \y 

Az        A./'  +  lAy  A./'  +  /Ay  ^'         ^~   ■^ 

where  the  derivatives  are  formed  for  (n,  ],)  and  where  I  is  an  infinitesi- 
mal complex  number.  AVhen  \z  approaches  0,  both  A./'  and  Ay  must 
approach  0  without  any  implied  relation  between  them.  In  general  the 
limit  of  \F/\z  is  a  double  limit  (§  44)  and  may  therefore  depend  on 
the  way  in  wliicdi  A.'-  and  Ay  approach  their  limit  0. 

XoAV  if  first  Ay  =  0  and  then  subsequently  A,/-  =  0.  the  value  of  the 
limit  of  AF/Ar;  is  A'-,'.  +  lY'^  taken  at  the  point  (",  //j  ;  wliereas  if  first 
A^'  =  0  and  then  Ay  =  0,  tlie  value  is  —  IX'^  -\-  Y'^.  Hence  if  the  limit 
of  AF/Az  is  to  lie  indepeiident  of  the  way  in  which  A."  approaches  0,  it 
is  sui'ely  necessary  that 

£X        .££__.  rX       ar 
ex  c.c  cy        cy 

cX       cY  cX  cY 

or  -7—  =  -—     and     —  =  —  -;—•  ^22) 

c.i'        cy  cy  CX  ^     ' 

And  conversely  if  tliese  relations  are  satisfied,  tlum 

AF       (cX   ,    .cY\       ^       icY       .cX\ 
A.v         \C.r  CX  J  \cy  cy) 

and  the  limit  is  A''  +  lY'^  =  Y',  —  iXy  tak(^n  at  tlie  ]>oint  (n,  //),  and  is 
independent  of  the  way  in  which  As  approaches  zei-o.  The  desirability 
of  having  at  least  tlie  ordinary  functions  differentiabh^  suggests  the 
definition:  A  complex  funrfuni  F(^x,  y)=Xi.r,  y)-\-(Y{x.  y)  is  rnn- 
s'uhrred  as  a  function  of  tlie  conijjlex  va ruihle  z  =  .'•  -\-  iy  irlicn  (in(J  only 
vJicn  X  and  Y  are  in  yenprol  (Jiffercnfiiihle  and  satisfy  the  rtdntions  (22). 
Til  tliis  cnsc  t///'  dri'irotire  is 


COMPLEX   XL^^[BEKS   AND  VECTORS  1.39 

,     ,        ilF       cX        .cY      cY       .cX  ,^„ 

^  ^        dz        CX  CX        (^il  ^y 

These  conditions  may  also  be  expressed  in  polar  coordinates  (Ex.  2). 

A  few  words  about  tlie  function  <I>(.r,  y).  This  is  a  multiple  valued  function  of 
the  variables  (x,  y),  and  the  difference  between  two  neiu-libdrin^:  branches  is  the  con- 
stant 2  7r.  The  application  (if  the  discussion  of  §  45  to  this  case  shows  at  once  that, 
in  any  simply  connected  reizion  of  the  complex  plane  which  contains  no  point  («,  li) 
svich  that  R{n,  h)  =  0,  the  different  branches  of  $(.c,  y)  may  be  entirely  separated 
so  that  the  value  of  $  nuist  return  to  its  initial  value  when  any  closed  curve  is  de- 
scribed by  the  point  (.f.  y).  If.  however,  the  region  is  multiply  comiected  or  contains 
points  for  which  U  —  0  (which  makes  the  reuion  nudtiply  cimni.'cteil  because  these 
points  nmst  be  cut  our),  it  may  happen  that  there  will  be  circuits  fur  which  <t>, 
although  changing  contiuuuusly.  will  imt  return  tn  its  initial  \alue.  Indeed  if  it  can 
be  shown  that  <i>  dues  imt  ivtui'ii  to  its  initial  value  when  cJiauging  continuously  as 
(.r,  y)  describes  the  boumlary  of  a  region  simply  comiected  except  for  the  excised 
points,  it  may  be  inferred  that  there  must  be  points  in  the  region  fi_)r  which  R  =  0. 

An  applicatinn  uf  tliese  results  may  be  made  to  give  a  very  simple  demonstration 
of  Uie  fundamental  tltcorcm  of  nlgebra  that  every  equation  of  the  nth  degree  lias  at  least 
one  root.    Consider  the  fuurtidu 

F{z)  =  z"  +  a,r'-i  +  •  •  ■  +  <i„-iz  +  a„  =  X(.r.  y)  +  iY(.r.  y), 

where  A' and  i'  ai'e  found  by  writing  z  as  ,r  +  ly  and  expanding  and  rearranging. 
The  functions  X  and  Y  will  be  polynomials  in  (./;.  y)  and  will  therefore  be  everj"- 
where  finite  and  continuous  in  (x,  //).    Consider  the  angle  <I>  of  F.    Then 

*  =  ang.  of  F  =  an-:,  of  a"  ( 1  +  —  H h  ^'"  "-  +  —  )  =  an-,  of  2"  +  ang.  of  (1  +  •  •  •). 

\         z  2''-i       z"l 

Next  draw  about  the  origin  a  cii'ele  of  radius  r  so  large  that 

I",' 


+  ••■  +  1^^-  '  + 


Then  for  all  points  z  upon  tlie  circumference  the  angle  of  F  is 

<J>  =  ang.  of  F  =  ?( (ang.  of  z)  +  ang.  of  (1  -|-  7;),         \'n\<^- 

Now  let  the  point  (.r,  y)  describe  the  circumference.  The  angle  of  z  will  change  by 
2  TT  for  the  complete  circuit.  Hence  <J)  must  change  In'  2mr  and  does  not  return  to 
its  initial  value.  Hence  there  is  within  the  circli'  at  least  one  point  (a.  h)  for  which 
li{(i.  h)  ~  Oand  conseouently  for  which  X{a,  h)  =  0  and  Y {a.  h)  ~  0  and  F{a,  b)=0. 
Thus  if  a  =  (I  +  ih.  then  /•"(<» )  =  0  and  the  ecjuation  F(z)  =  0  is  seen  to  have  at 
least  the  one  root  a.  It  follow^  that  z  —  a  is  a  factor  of  F [z)  ;  and  hence  by  induc- 
tion it  may  be  seen  that  F(z)  =  0  has  just  n  roots. 

74.  The  discussion  of  tlie  algelmi  of  complex  numbers  showed  how 
the  sum,  difference,  jiroduct,  quotient,  real  powers,  and  real  roots  of 
such  numl)ers  could  ho  found,  and  hence  made  it  possil)le  to  compute 
the  vabie  of  any  ^^'ivcn  alg'oliraic  expression  oi-  function  of  ,-;  for  a  y-iveii 
value  of  ;;.    It  renuiins  t<.)  sIkjw  that  any  alj^^'braic  expression  in  z   is 


160  DIFFERENTIAL   CALCULUS 

really  a  function  of  z  in  the  sense  that  it  has  a  derivative  with  respect 
to  z,  and  to  find  the  derivative.  Xow  the  differentiation  of  an  algebraic 
function  of  the  variable  .r  was  made  to  depend  upon  the  formulas  of  dif- 
ferentiation, (6)  and  (7)  of  §  2.  A  glance  at  the  methods  of  derivation 
of  these  formulas  shows  that  tliey  were  proved  l)v  ordinary  algebraic 
manijoulations  such  as  have  Ijeen  seen  to  l)e  equally  possible  Avith  imagi- 
naries  as  with  reals.  It  therefore  may  be  concluded  that  (in  a  hj  eh  rule 
e.i'pression  In  z  /ms  (t  (hrrlrdflre  icifli  rrspt'ct  in  z  and  that  Oi'i'lruflrc 
VKn/  he  found  just  us   if  z  were  a   ri'dl  ru r'uthh'. 

The  case  of  the  elementary  functions  r,  log,-;,  sins;,  cos  ,v,  ■••  otlici- 
than  algebraic  is  different;  for  these  functions  have  not  been  defined 
for  complex  vai'iables.  Now  in  seeking  to  define  these  functions  when  :: 
is  complex,  an  effort  should  l^e  made  to  define  in  such  a  way  that:  1° 
when  z  is  real,  the  new  and  the  old  definitions  Ijecome  identical  :  and 
2°  the  rules  of  operation  with  the  function  shall  be  as  nearly  as  possi- 
ble the  same  for  the  complex  domain  as  for  the  real.  Thus  it  would  In- 
desirable  that  7>'^' =  r^  and  </ + "' =  e'f"',  when  z  and  /'•  are  complex. 
With  these  ideas  in  mind  one  niay  proceed  to  define  the  elementary 
functions  for  complex  arguments.    Let 

i~-  =  /.' (./■,  //)  [cos  ^ (,'■,  //)  -f  ;  sin  $(,/■,  //)].  (24) 

The  derivative  of  this  function  is,  by  tlie  iirst  rule  of  (23), 

c  c 

1J('~  =  -^  ( A'  cos  <!>)  4-  /  —  (  R  sm  4>) 
c.r  ■  C.r 

=  (/.','.  cos  <t>  —  7.'  sin  <P  ■  $,')  +  /  (  /,',',  sin  <$•  -f  A'  cos  <I>  •  $,',), 

and  if  this  is  to  be  identical  with  '-  ab(i\-c,  the  e(piations 

A',',  cos  <I>  —  A'4)'.  sin  $  =  Ji  cos  <^  A','  =  A' 

or 
A',',  sin  <I>  -f  L'<t>',.  cos  4>  =  /.'  sm  <P  f,,  =  0 

must  hold,  wliei'c  the  second  pair  is  obtained  by  solving  the  first.  If 
the  second  i'oi'ni  of  the  dci-ivative  in  (2.'))  had  licen  used,  the  results 
would  have  licen  /,'J  =  0.  <i>^'^  —  \.  It  tliei-efoi-e  appears  that  if  the 
derivative  of  r~.  however  computt'd,  is  to  be  i'-.  then 

7.';  =  n,    /,',;  =  o,    ci',:  =  o.    <p;,  =  i 

are  four  conditions  ini])0se(l  upon  /.'  and  (p.  These  conditions  will  be 
satisfied  if  7'  =  >■''  aiid  <I>  =  >/.*    Hence  define 


i'~  =  ('■ 

'■  ^ '■"  =  ,-'■  (COS //  -f- ; 

■  sin  //) 

*  The  use 

of  tlH 

■  iiiori 

!■  L^i'iirral 

Snlutiiills    /.'     -    ',■'',  'p 

,/   -L    1  • 

iiiicli  wdiili 

1  III  It  1- 

ClliU-L 

■  til  '■'   w i 

irii  1/       (1  ami  z       .'■  Ill- 

W.Mlia   I! 

'/  -i-  ''  wdiilil  Irail  to  (•xiir<'ssiciti> 


COMPLEX  NUMBERS  AXD  VECTORS  161 

With  this  definition  7>e'  is  surely  e^,  and  it  is  readily  shown  that  the 
exponential  law  e'  +  ""  =  (fe"'  holds. 

For  the  special  values  \  iri,  ir't,  2  irl  of  ,.  the  value  of  (f  is 

t'2"'  =  /,       r--  =  -l,       <--'  =  !. 

Hence  it  appears  that  if  2  inrl  he  added  to  z,  ('""  is  unchanged; 

t--  +  -"''  =  r",     period     2  tt/.  (26) 

Thus  m  tltp  rnriipli'.r  iJnnuiln  r^  Jid.-i  iln>  period  '2rrl.  just  as  cos./'  and 
sin  a-  have  the  real  period  2  tt.    Tliis  relation  is  inherent;   for 

e'"  =  cos  //  +  ''  sill  !/•     li''"  =  (^os  u  —  l  sin  //, 

''"'  +  ''"■"'         .  '■'"  —  ''"■'"  ,o_ 

and  cos  y  = . ;     sm  //  = — •  (2i  ) 

2  '  2  <. 

The  trigonometric  functions  of  a  real  variahle  //  may  l)e  cxjiressfd  in 
terms  of  the  exponentials  of  yl  and  —  ///.  As  the  exponential  has  Iteen 
defined  for  all  complex  values  of  .-.-,  it  is  natural  to  use  (27^  to  define 
the  trigonometric  functions  for  complex  values  as 

cos  z  =         ^,         '     sin  ,-:  =  — ~ (J .  ; 

With  tliese  definitions  the  ordinary  formulas  for  cos  (-;  +  "'),  ^>  sin  ,-.',  •  •  • 
may  be  obtained  and  Ije  seen  to  hold  for  com])lcx  ai'gunients,  just  as  tlie 
corresponding  formulas  Avere  derived  for  tlie  hy])erbolic  functions  (§  ~j). 
As  in  the  case  of  reals,  the  logarithm  log  ,v  will  be  defined  for  com- 
plex numbers  as  the  inverse  of  the  exponential.    Thus 

if  i'~  =  >r,     tlien     log  //•  =  ,v  +  2  niri,  (2S) 

where  the  periodicity  of  the  function  r~  sliows  tliat  fl/r  /of/r/rif/nn  is  not 
iini<iuehj  fh:t<;rtnini-d  hut  (idinlts.  flic  (iddlflnn  <if  'Inirl  in  (niij  one  <>f  !fs 
values,  just  as  tan~^  .r  admits  tlie  addition  f)f  ?i7r.  If  ir  is  written  as  a 
complex  numl)er  ii  +  ir  with  modulus  /•  =  '^'  ir  +  c'-  and  Avith  the  angle 
</).  it  follows  that 

//•  =.  „  +  '■'•  =  '•('•OS  (/)  +  /  sin  <^)  =  rr'-''  =  ,>-'--^'J"  :  (29) 

and  log  //•  —  l(.)g  /•  +  ^/  =  log  A  li-  +  r-  +  i  tail"'  ('■/") 

is  the  expression  for  the  logarithm  of  //•  in  terms  of  t]n.^  hkhIuIus  and 
angle  of  ?/■ ;  tin.'  '2  rnri  mav  he  added  if  desired. 

To  this  point  tlie  expression  of  a  power  a^'.  Avhere  the  ex])()ne]it  //  is 
imaginary,  has  had  no  definition.  The  definition  niay  now  be  givoi  in 
terms  of  exponentials  and  logarithms.    Let 


162 


DIFFERENTIAL   CALCULUS 


In  this  "vvcay  the  problem  of  computing  c^  is  reduced  to  one  ah-eady 
solved.  I'roni  the  very  definition  it  is  seen  that  the  logarithm  of  a 
power  is  the  product  of  the  exponent  hy  the  logai'itlnn  of  the  base,  as 
in  the  case  of  reals.  To  indicate  the  path  that  has  been  followed  in 
defining  functions,  a  sort  of  family  tree  may  be  made. 


real  numl)ers,  .c 

I 
real  powers  and 
roots  of  reals,  x" 

I  I 


real  angles,  x 

\" 
real  trigonoiueti-ic  functions, 
cos  J,  sin/,  tan-ij,  •  •  • 


exponentials,  logarithms 
of  reals,  t-''.  \o<ix 


real  powers  and  rex  its 
of  iniairinaries.  z" 


I  I 

exponentials  of  iniauinaries,  e^ 


logarithms  of  imaginaries,  lo<j 
imaginary  powers,  z" 


trigonometric  f mictions 
of  imasiinaries 


EXERCISES 


1.  Show  that  the  following  complex  functions  satisfy  the  conditions  (22)  and 
are  therefore  functions  of  the  complex  variable  z.    Find  F'(z): 


(5)  log  Vj:-  +  >f  +  i  tan- 1  -  , 
(f)   sin  J-  sinh  y  +  i  cos/  cosh  //. 


(a)  X-  -  ij-  +  2  ixy, 

(7)  -r-^-,  -  '  ;o-^ 
( e )  &'  cos  y  +  ie'  sin  //, 

2.  Show  that  in  polar  ciHirdinates  the  cimditions  for  the  existence  of  F'{z)  ar 
cX  _lcY_       rT 

cr        r  c<p         cr 

3.  Use  the  conditions  (if  Ex.  2  to  show  from  J)  lo 


1^     with     F'(.)  =  (^+^^)(n,s^ 
r  cd>  \  cr 


I  sm 


z-  1  that  log  z  =  log  r  +  (pi. 

4.  From  the  definitions  given  above  prove  the  formulas 

(or)    sin  (.r  +  ///)  =  sin  x  cosh  //  +  i  cos  x  sinh  //, 

(/3)   cos  (.r  +  iy)  =  cos  x  cosh  //  —  <  sin  j  sinh  y, 

sin2.r  +  /sinh  2// 

7     tan  (.r  +  ly)  = --—  • 

cos  2/  +  cosh  2  // 

5.  Find  to  three  decimals  the  complex  numbers  which  express  the  values  of: 
(e)  sin  \  TTi. 

(0  i-^'(-i). 


(.0   CO.- 


(7)  e-^  '  ^  ^     •',     (0) 

(t?)  sin(^  +  i  x- o).  C?)  taii(-l— /). 

(\)    l.igU+^x'^).  (a)    ln-(-l-;). 


6.  Owing  to  the  fact  tliar  logr;  is  multiple  valued.  */''  is  multiple  valued  in  such 
a  manner  that  any  one  value  may  be  nmltiplied  liy  c-"~'"'.  Find  one  value  of  each 
of  the  following  and  several  values  nf  one  of  them: 

(a)  2',         {13)  i'\         (7)^^.         (5)^,         (^)  (1  +  i^'-S)'''  '    ■ 


COMPLEX  XU3IBER8   AND   VECTORS 

7.  Show  tliat  Z)«^  =  a~  log  a  when  a  and  z  are  complex. 


163 


8.  Show  that  Oi''Y  =  a'"':  and  fill  in  such  other  steps  as  may  l)e  su,a-,i(ested  by 
the  work  in  the  text,  which  for  the  most  pjart  has  merely  been  sketched  in  a  broad 
way. 

9.  Sliow  that  \i  f{z)  and  g  {z)  are  two  functions  of  a  complex  variable,  then 
/(■z)±y(2),  ocf{z)  witli  a  a  complex  constant, /(z)^(z).  f{z)/(j{z)  are  also  func- 
tions of  z. 

10.  <)btain   louarithmic   expressions  for  the   inverse  triironometric   functicjns. 
Finil  sin- '/. 

75.  Vector  sums  and  products.  As  stated  in  §  71,  a  veotor  is  a  quan- 
tity wliich  lias  mag-nitudo  and  direction.  If  the  magnitudes  of  two 
vectors  are  ecjual  and  the  directions  of  tlie  two  vectors  are  the  same, 
tlie  vectors  are  said  to  be  (^(jual  irres})ective  (jf  tlie 
position  which  tliey  occu})}'  in  s})ace.  Tlie  vector 
—  a  is  hy  definition  a  vector  ■\vln(di  has  the  same  f^i 
magnitude  as  a  hut  tlie  0}ii)Ositc  direction.  Tlie  ^ 
vector  i/ia  is  a  voctoi-  which  has  the  sami'  diroction 
as  a  (or  the  oii])Osit(')  and  is  ///  for  —  /// )  times  as 
long.  The  law  of  vector  or  geonicti-ic  addition  is 
the  parallelogram  or  triangle  law  (§  71)  and  is  still 
ap])licable  Avlieii  the  vectors  do  not  lie  in  a  plane 
hut  have  any  directions  in  s])ace:  for  any  two  vec- 
tors brouglit  end  to  end  determine  a  plane  in  which  the  construction 
may  he  carried  out.  \'eetors  will  he  designat<'d  by  (4 reek  small  lettiu-s 
or  by  letters  in  heavy  type.  The  i-idations  of  e(piality  oi'  similaritv 
between  triangles  establish  the  rules 

a  +  {3  =  (S  +  a.   a-\-{(3-\-y)  =  in  +  ^ »  -f-  y.    ///  (a  -f  /3 )  =  i/<'t  +  ////?    ( ;;-50) 

as  true  for  vectors  as  well  as  for  numbei-s  whether  real  or  complex.    A 
vector  is  said  to  he  zero  when  its  magnitude  is  zero,  and  it  is  writ- 
ten 0.    Eroni  the  definition  of  addition  it  follows  that 
or  +  0  =  fi.    In  fact  us  fur  us  inlil'dion,  suhirai-fuin,  a  ad 
iniiltiiilii->ifl(,n   III/  nil iiiJicrs  nn-  nnicfnii'il,  n-rturs  nhi'ij 
fill'  s"//ii'    f'lriiiiil  Imrs  IIS  nil iiihrrs. 

A  vector  p  may  be  resol\'ed  int(^  components  }iar- 
allel  to  any  three  given  N'ectors  a.  ft.  y  wdiich  are  not 
parallel  to  any  one  ])lane.  For  let  a  jjaralleleiiiped 
be  consti'ucted  with  its  edges  ])ai-all(d  to  the  three 
given  vectors  and  with  its  diagonal  etpial  to  the  vectoi'  whose  compo- 
nents are  desired.     Tlie  edges   of  the  paralhdepiped  are   then  certain 


164  DIFFEEEXTIAL  CALCULUS 

inulti})les  xa,  y(3,  .ty  of  a,  (i.  y ;  and  these  are  the  desired  components 
of  p.    Tlie  vectoi-  p  may  be  written  as 

p  =  xa  +  ///3  +  .-y.*  (31) 

It  is  clear  that  two  equal  vectors  would  lun-essarily  have  the  same 
components  along  three  given  directions  and  that  the  components  of  a 
zero  vector  would  all  be  zero.  -Just  as  the  ecjuality  of  two  complex 
numbers  involved  the  two  equalities  of  the  resijective  real  and  imagi- 
nary })arts,  so  tlie  ecpiality  of  two  vectoi-s  as 

p  =  :ra  +  >//3  +  -:y  =  -r'n  +  //'/?  +  r:'y  =  p'  (31') 

involves  the  three  eipiations  ./•  =  .'•',  //  =  //,  r:  =  z'. 

As  a  problem  in  the  use  of  vectors  let  there  be  triven  the  three  vectors  a,  /3,  y 
from  an  assumed  oritiiu  0  to  three  vertices  of  a  parallelogram  ;  refjuired  the  vector 
to  the  other  vertex,  the  vector  expressions  for  the  sides  and  diagonals  of  the  paral- 
lelogram, and  the  proof  of  the  fact  that  the  diagonals  bisect 
each  other.  Consider  the  iigure.  Tlie  side  A  IS  is.  by  the 
triangle  law,  that  vector  which  when  added  to  OA  =  a 
gives  OB  =  ^.  and  hence  it  nnist  l)e  that  Ali  =  (i  —  ex. 
In  like  manner  .K'  =  7—  a.  Now  OD  is  the  sum  nf  OC 
and  CI),  and  CI)  =  AH:  hence  01)  =  7  +  /3  -  a.  The  diag- 
onal AIJ  is  the  difference  of  the  vectors  OD  and  OA.  and 
is  therefore  y  +  (3  —  2  a.  The  diagonal  1>C  is  7  —  /3.  Now  the  vector  from  0  to  the 
middle  point  of  ]IC  may  be  found  Ijy  adding  to  OB  one  half  of  BC.  Hence  this 
vector  is  /:i  -t-  4  (7  —  /i)  <)r  \  (^  +  7).  In  like  manner  the  vector  to  the  middle  point  of 
AD  is  seen  to  be  a  +  H.7  +  /3  —  -  <»)  "r  I  (7  4-  /3).  which  is  identical  with  the  former. 
The  two  middle  points  tlierefore  coincide  and  the  diagonals  bisect  each  other. 

Let  (I  and  (3  be  any  two  vectors,  \a\  and  \(3\  their  respective  lengths, 
and  /I  (a.  jS)  the  angle  between  them.  For  convenience  the  vectors  may 
be  considered  to  be  laid  otf  from  the  same  origin.  The  product  of  the 
lengths  of  the  vectors  by  the  cosine  of  the  angle  lietween  the  vectors 
is  called  the  sra/af  j,fot/iN-f, 

scalar  product  =  a.^  =  |^!!^1  ''^^^  ^  <«•  /?),  (32) 

of  the  two  vectors  and  is  denoted  l)y  jilaeing  a  d(.»t  between  the  letters. 
This  combination,  called  the  scalar  ]troduct,  is  a  number,  not  a  vector. 
As  1^1  cos  z^  (a'.  /3)  is  the  jirojeetion  of  /?  u])on  the  direction  of  a,  the 
scalar  ])ro(luet  may  be  stated  to  lie  equal  to  the  prodtict  of  the  length 
of  eitlier  vector  l>y  the  lengtli  of  the  iirojcction  of  the  other  iqton  it. 
Ill  jiarticular  if  either  \'ector  were  of  miit  length,  the  scalar  jiroduct 
would  bt'  the  projection  of  tlie  (jtlier  upon  it.  with  projter  regard   for 

*  The  numbers  ./•.  >/.  z  are  tlic  (ibli(|nr  coru'diiiatt's  of  tlie  torniiiia!  end  ai  p  (if  the 
initial  cnrl  be  at  the  (iriudii)  i-cffi-icil  to  a  scr  of  axes  whicli  are  i)arallcl  to  a.  pS.  7  and 
upon  winch  the  unit  lengths  are  taken  as  the  lengths  of  n,  /3,  7  respectively. 


COMPLEX  Xl'.Mr,ErvS   AND  YECTOKS  IGo 

the  sign  ;  and  if  Ijotli  vec-t(«'s  arc  unit  vectors,  tlie  pi'oduct  is  the  cosine 
of  the  angle  between  them. 

The  scalar  product,  from  its  delinition,  is  (•(immutdtire  so  i\vAt(X'^=^'<x. 
Moreover  (/tia)'fS  =  a'(ii)(3)  =  m  {<X'(i),  tlius  allowing  a  numerical  factor 
m  to  be  combined  with  either  factor  of  the  product.  Furthermore  the 
distrlhatirc'  hnr 

a.(i3  +  y)  =  n-/3  +  n.y      or      (a  +  ^).y  =  a.y  +  yS-y  (33) 

is  satisfied  as  in  tlie  case  of  nuiubcrs.  Foi-  if  a  be  written  as  tlie  product 
(/a^  of  its  length  /t  by  a  vector  a^  ol'  unit  length  in  tlie  direction  of  (x, 
the  first  equation  ])ecomes 

(la^'ift  +  y)  —  '"•'^i*/?  +  "(t^'y     or     (x^-([i  +  y)  =  a^'(i  +  rc^-y. 

And  now  «j»(/3  +  y)  i^  the  |)rojection  of  tlie  sum  /?  +  y  n})on  the  direc- 
tion of  a,  and  cc^'P  +  'tj.y  is  tlie  sum  of  the  ])i-ojections  of  /Sand  y  iqion 
this  direction;  liy  the  law  of  projections  these  are  equal  and  hence  the 
distributive  law  is  })roved. 

The  associative  law  does  not  hold  for  scalar  products  ;  for  («•/?)  y 
means  that  the  vector  y  is  multijilied  by  the  numVjer  a-^,  Avliereas 
«  (y8«y)  means  that  a  is  multiplied  by  (/3»y),  a  very  different  matter. 
The  laws  of  cancellation  cannot  hold:   for  if 

«./?  =  0,     then     [a"/3|  cos  Z  (a.  /?)  =  0,  (34) 

and  the  vanishing  of  the  scalar  pi'oduct  a-ft  implies  either  that  one  of 
the  factors  is  0  or  that  the  two  vectors  are  ])er])endicular.  In  fact 
a.yg  =  0  is  called  the  cnndlfuni  of  ix'rpcn'ilciild r'lt ij.  It  should  be  noted, 
however,  that  if  a  vectoi'  p  satisfies 

p.a  =  0,         p./3  =  0,         p.y  =  0,  (35) 

three   conditions   of   perpendicularity  with   three   vectors   a,  /3.  y   not 
parallel  to  the  same  ])lane.  the  inference  is  that  p  =  0. 
76.   Anotlier  product  of  two  vectors  is  the  rfcfnr  prod  net  ^ 

vector  product  =  ax/3  =  v  a'}  fi\^n\  A  ((t,  ft),  (oG) 

where  v  represents  a  vector  of  unit  length  normal  to  the  plane  of  a 
and  /3  upon  that  side  on  which  I'otntioii  from  a  t(j 
yS  through  an  angle  of  less  than  1S()°  apiu-ns  posi-  axiSt 
five  or  counterclockwise.  Thus  the  vector  ]>roduct 
is  itself  a  vector  of  which  the  direction  is  ])er])en- 
dicular  to  each  factor,  and  of  whicli  the  magni- 
tude   is    the    ])roduct   of   the    magnitudes    into   the 

sine  of  the  included  angle.  The  magnitude  is  therefoi'e  equal  to  the 
area  of  the  parallelogram  of  which  the  vectors  a  and  /3  are  the  sides. 


IGG  DIFFERENTIAL   CALCULUS 

The  vector  })roduct  will  be  represented  Ijy  a  cross  inserted  between  the 
letters. 

As  rotation  from  ^  to  a  is  tlie  oppositt;  of  tluit  fr(jni  a  to  /3,  it  follows 
from  the  definition  of  the  vectcjr  product  that 

fixa  =  -  ux/3.      not      ax^  =  /3xa-,  (37) 

and  the  product  is  not  cdhhh  iif<ifii-i'.  the  order  of  the  factors  must  be 
carefully  observed.    Furthci-morL'  the  e(piation 

ax  13  =  V  a'\l3'<s\nA(a,  /3)  =  0  (38) 

implies  either  that  one  of  the  factors  \;uiishes  or  tliat  the  vectors  ft  and 
/3  are  ])arallel.  Indeed  the  condition  ax/3  =  0  is  called  the  cnndiflnn  nf 
piirdlli'l'tsiii.  The  laws  of  cancellation  do  not  hold.  Tlie  associati^■e  law 
also  (l(_)es  n(jt  hold;  for  (axft)xy  is  a  vector  jierjiendicular  to  ax/3  ;inil  y. 
and  since  ax^  is  perpendicular  to  the  plane  of  tc  and  (3.  tlie  vector  (ax/3)xy 
perpendicular  to  it  must  lie  in  the  plane  of  a  and  /3 :  whereas  the  vec- 
tor ax((3xy),  by  similar  I'easoniny,  must  lie  in  the  jilane  of  /3  and  y ;  and 
hence  tlie  two  vectors  cannot  be  (^qnal  exi-ept  in  the  very  special  case 
where  each  was  ])arallel  to  f3  which  is  comnion  to  the  two  ])lanes. 

lUit  the  opei'ation  ( //ki  ^y /3  =  nx(  i//f3}  =  ///(ax/Ji.  which  consists  in 
alhnving  the  transference  of  a  numerical  factor  to  any  position  in  the 
])roduct,  does  hold;   and  so  does  the  (llsfriliiifii-c  Jan- 

«:x(/3  -f  y)   =  ax/S  +  axy        and        (  a  +  /3)xy  =  axy  -f-  ^xy.  (.SO) 

the  ])r(jof  of  which  will  lie  L^'iveii  l.telow.  In  ex]iaiidiii,L;'  accoi-diiiL;-  to 
the  distributive  law  care  must  be  exercised  t(j  kee])  tlie  order  <.»f  tlie 
fa(;tors  in  each  vector  product  the  sanu^  on  both  sides  of  the  eipiation. 
owing  to  the  failure  of  the  commntative  law:  an  interchan,!-;'e  (.)f  the 
order  of  tlie  factors  chanyes  the  si^'ii.  It  might  seem  as  if  any  algeliraic 
operations  where  s(_)  many  of  the  laws  of  elementary  algebra  fail  as  in 
the  case  of  vector  ])roducts  would  be  too  restricted  to  be  \vv\  useful; 
that  this  is  not  so  is  due  to  the  astonishingh"  great  number  of  pi'obh^ms 
in  whicdr  the  analysis  can  be  cai'i'ied  on  with  only  the  laws  of  addition 
and  the  distributive  law  of  multiplication  conibined  with  the  jiossibility 
of  transferring  a  nuniei-ical  factor  from  one  position  l<i  another  in  a 
jiroduct;  in  addition  to  these  laws,  the  scalar  ])roduet  cr»/3  is  eommuta- 
tive  and  the  vector  jiroduct  'tx/3  is  commutative  except  bir  chaiigi^  of  sign. 
In  addition  to  segments  of  liiu^s.  jilnnr  (irms  tiKuj  lie  ri'ijn nli-il  ".-■ 
i-fffor  ijiinnfiflfs  :  for  a  ]ilane  area  has  magnitude  (the  amount  of  the 
area)  and  direction  (the  dii'ection  of  the  noinial  to  its  ]ilane ).  To  S]tecify 
on  which  side  of  the  ]ilane  the  normal  lies,  some  con\'ention  must  be 
made.    If  the  ai'ea  is  part  oi'  a  surface  inclosing  a  jiortion  of  Sjiace.  the 


COMPLEX  XU:\IBERS   AXD   VECTORS 


1(3' 


A^ 


normal  is  taken  as  the  exterior  normal.    It'  the  area  lies  in  an  isolated 

plane,  its  positive  side  is  determined  only  in  connection  with  some 

assigned  direction  of  description  of  its  hounding  curve;  the  rule  is  :   If 

a  person  is  assumed  to   walk  along  the  boundary  of  an  area  in  an 

assigned  direction  and  upon  that  side  of  the  plane  which 

causes  the  inclosed  area  to  lie  upon  his  left,  he  is  said 

to  be  upon  the  positive  side  (for  the  assigned  direction 

of  description  of  the   boundary),  and  the  vector  which 

represents  the  area  is  the  normal   to  that  side.    It  has 

been   mentioned    that    the    vector    ])roduct    represented 

an  area. 

That  the  projection  of  a  plane  area  u})on  a  given  plane  gives  an  area 
which  is  the  original  area  midtiplied  by  the  cosine  of  the  angle  between 
the  two  planes  is  a  fundamental  fact  of  projection,  following  from  the 
simple  fact  that  lines  parallel  to  the  intersection  of  the  two  planes  are 
unchanged  in  length  whereas  lines  perpendicular  to  the  intersection 
are  multiplied  by  the  cosine  of  the  angle  between  the  planes.  As  the 
angle  l)etween  the  normals  is  the  same  as  that  Ijetween  the  planes,  f/te 
projrctlini  of  an  (iraa  ^(pon  a  plane  and  tin'  jtrnjcctidii  of  flic  ci'ctor  rep- 
resenting the  (ireii  upon  the  nurnidl  to  the  jdmie  are  cqulrdlent.  The 
projection  of  a  closed  area  upon  a  plane  is  zero;  for  the  area  in  the 
projection  is  c(n'ered  twice  (or  an  even  numljcr  of  times)  witli  opposite 
signs  and  the  total  algebraic  sum  is  therefore  0. 

To  prove  the  law  ax(/3  -)-  y)  =  av.^  -(-  axy  und  illustrate  the  use  of 
the  vector  interjjretation  of  ai-eas,  construct  a  triangular  piism  with  the 
triangle  on  /3,  y.  and  /3  +  y  as  base  and  a  as  lateral  edg(\  The  total 
vector  expression  for  the  surface  of  this  ])rism  is 

y3x(t  +  yx<t  +  ax(/3  +  y)  +  .V(/3xy)  -  \  /3xy  =  0, 

and  vanislies  because  the  surface  is  clos(-d.  A  cancel- 
lation of  the  eijual  and  o})positc  terms  (the  tw(j 
leases)  and  a  simple  transposition  combined  with  the 
rule  /3xct  =  —  ax^  gives  the  result 

frx(/3  -(-  y)  =  —  i^^"^  —  y^"-'  =  '^'^/S  +  axy. 

A  svstem  of  rectors  of  reference  wliicli  is  parti(,-ularly  iiseful  consists 
of  three  vect(jrs  i,  j,  k  of  unit  length  directed  along  the  axes  A',  1',  Z 
drawn  so  that  rotation  from  A'  to  }'  a})pears  positive  from  the  side  of 
the  ./'//-plane  u})on  which  Z  lies.  The  com])(jnents  of  any  \-cctor  r  drawn 
from  the  origin  to  tlit^  jioint  (',/■,  //,  ,-;")  are 

,'/i.      //J.      ,-;k,      and      r  =  ./i  +  //j  +  .-.k. 


168  DIFFERENTIAL  CALCULUS 

The  products  of  i,  j,  k  into  cucli  otlicv  arc,  from  t\w  definitions, 

i-i  =  j-j  =  k.k  =  l, 
i-j  =  j-i  =  j-k  =  k.j  =  k.i  =  i.k  =  0, 

ixi  =  jx  j  =  kxk  =  0,  ^     ^ 

ixj  =  —  jxi  =  k,     jxk  =  —  kxj  =  i,     kxi  =  —  ixk  =  j. 

By  means  of  tliese  prodm^ts  and  the  distributive  laws  for  scahir  and 
vector  produ(!ts,  any  giv(Mi  products  may  be  expanded.    Thus  if 

a  =  ^^^i  +  aj  +  (1,^     and     fi  =  A^i  +  />j  +  ^'Js., 

then  a-.^  =  a/>^  +  r/./.,  +  a.h,^,  (41) 

ay.  13  =  (nj>^  -  r//gi  +  ("■/>,  -  "/',)  j  +  ("/',  -  ''7'i)k, 

by  direct  multiplication.    In  this  Avay  a  passage  may  b(^  made  from 
vector  formulas  to  Cartesian  formulas  whenever  desired. 

EXERCISES 

1.  I'rovc  ^coiiictricall.y  tluit  (t:  +  (p  +  7)  =  (<i'  +  (i)  +  y  and  11/ {it  +  /i)  =  111a  +  iu(i. 

2.  ]f  a  and  ft  arc  tlui  vectors  from  an  assumed  oi-ii;in  to  A  and  li  and  if  (' 
divides  All  \\\  the  ratio  m  :  n,  sliow  that  tlie  vector  t.o  (,'  is  7  =  [ita  +  iuji)/{)n  +  it). 

3.  In  tlie  ])ara,lleh>,uram  AIICI)  show  tliat  the  line  HE  coiniectinii,-  tlie  vertex  to 
the,  middle  point  of  tlu^  oi^posite  sidi^  CD  is  triset'ted  Ijy  the  diai^'onal  AJ)  and 
trisects  it. 

4.  Show  that  tin;  medians  of  a  triangle  meet  in  a,  point  and  are  trisected. 

5.  ]f  *;/[  and  iu.,  are  two  masses  situated  at  7'^  and  7*.,,  the  rotter  of  (jraviiy  or 
rvnlif  0/  >n<(ss  of  m^  aud  111.,  is  dclined  as  that  point,  (!  ou  tlie  line  l\f'.,  which 
di\ides  /',/'.,  iu\<'rsely  as  the  masses.  Moreos'er  if  CV,  is  the  center  of  mass  of  a 
nundier  of  masses  of  whi(di  the  total  luass  is  .1/,  and  if  (,'.,  is  tlie  ceiilef  of  mass  of 
a  niuuher  of  other  masses  whose  total  mass  is  .)[.,,  tin."  same  rule  applied  to  M^  aud 
37„  and  C/,  and  ^/.,  !.;ives  the  center  of  gravity  (,'  of  the  total  number  of  masses. 
Show  that, 

in.r.  +  m.,r.,  ,  ?/;,r,  +  ?//.x,  -»-■••  +  ?H„r„      Sw^r 

r  ^      '   '  -  -     aud     r  — --'-  ' -  -   ■   -  :-- -    --, 

'"i  +  "i.j  "'1  +  "'o  +  •  •  ■  +  '"„  i:»(, 

whci'c  r  denotes  the  vector  to  the  center  of  ni'avily.     liesolve  inlo  comjionents  to 
.show  ,,  „  .. 

.1-  r:-       —    ,        //  :.:  -'    ,        Z  : -.  - 

6.  if  a  and  ji  are  t.wo  Jixed  vectors  aud  p  a  \ariahh>  xector.  all  lieiiti;- laid  off 
from  the  same  oriniii.  show  that  (p  —  ft)-"-  =  •*  i''^  tht'  equation  of  a,  plane  llii'ounh 
the  cud  of  li  ])ei-i)eu(lieular  to  (f. 

7.  Let,  cc,  (i.  7  he  the  vectors  to  the  vertices  .1.  J!,  ('  of  a  trian-le.  W'l'ite  Ihe 
thiee  e(|ualions  of  Ihe  planes  through  the  vertices  ]>erpeuilicular  to  the  o]ii)osite 
sides.  Show  that,  the  third  of  these  can  he  derived  as  a.  comliinatiou  of  the  other 
two;  and  hence  iufer  that  the  three  ]ilaiies  have  a  line  in  common  and  tiiat  the 
perpendiculars  from  the  vertices  of  a  triait^le  meet  in  a,  point. 


COMPLEX  NUMBERS  AXD  VECTOES  1G9 

8.  Solve  the  xn-oblem  analogous  to  Ex.  7  fur  the  perpendicular  bi.sectoi-.s  of  the 
sides. 

9.  Note  that  the  length  of  a  vector  is  v  axr.    If  a,  p,  and  y  =  (3  —  a  are  the 
three  sides  of  a  triangle,  expand  7.7  =  (/i  —  a)'{l3  —  a)  to  obtain  the  law  of  cosines. 

10.  Sliow  that  the  sum  of  the  scjuares  of  the  diagonals  of  a  parallelogram  ei|uals 
the  sum  of  the  scjuares  of  the  sides.  What  docs  the  dilfcrence  of  the  sqiiares  of  the 
diagonals  cental  ? 

(X*8  ( (Xy.  8\  X  (f 

11.  Show  that  ^     (f  and      -  are  the  conix)onents  of /i  parallel  and  iierpcu- 

d'ct  cixr 

dicular  to  a  by  showing  1°  that  these  vectors  have  the  right  direction,  and  2°  tiiat 
they  have  the  right  magnitude. 

12.  If  (f,  13,  7  arc  the  three  edges  of  a  xiarallelepiped  which  start  from  the  same 
vertex,  show  that  (ax/i).7  is  the  volume  of  the  parallelepi})cd,  the  volume  being 
considered  positive  if  7  lies  on  the  same  side  of  the  plane  of  ci  and  (3  with  the 
vector  axfj. 

13.  Show  by  Ex.  12  that  (<tx/i).7  =  a'{l3xy)  and  {(ixl3)'y  —  {l3xy)'a  ;  and  hence 
infer  that  in  a  product  of  three  vectors  with  cross  and  dnt.  the  x'ositidu  of  the  cross 
and  dot  may  l)e  interchanged  and  the  order  of  the  factors  may  be  permuted  cyc- 
lically without  altering  the  value.  Show  that  the  vanishing  of  {i\xj3).y  (U-  any  of 
its  equivalent  expressions  denotes  that  a,  (3,  7  are  parallel  to  the  .same  plane  ;  the 
condition  a'x/3«7  =  0  is  called  the  condition  of  complanarity. 

14.  Assuming    a  =  «ji  +  a.,}  +  r(..k,    l3  =  li^i  +  ^,j  +  ^gk,    7  —  c,i  -|-  r.,j  +  ('..k, 
expand  (r'7,  a«/S,  and  (ix{i3xy)  in  terms  of  the  coelticients  t(j  show- 
ax  (/ix7)  =  {(fy)ft  —  (((-•/i)7;     and  hence     (ax/i)x7  =  (^cfy) fi  —  {y'i3)a. 

15.  The  formulas  of  Ex.  14  for  expanding  a  i)rodnct  with  two  crosses  and  the 
ride  of  Ex.  13  tliat  a.  dot  and  a  cross  may  be  interchanged  may  be  ajiplicd  to  expand 

(ax/i)x(7x5)  =  (<i.7x5),:;  -  (,i-7x5)a  ..  (cix/^.5)7  -  {^i-,i-y)5 
and  {axi3). {yx5)  ~^  {a.y)il3.5)  -  {i3.y)i<r.5). 

16.  If  a  and  f3  are  two  unit  vectors  in  the  ./■//-phine  inclined  at  angli's  ff  and  (f> 
to  the  .f-axis,  show  that 

a  =  ic'os^  +  j  sin  0,     fi  =  icos<;6  +  jsin  4> ; 

and  from  the  fact  that  (t./i  =  cos  (0  —  6)  and  ux/i  =:  ksin(0  —  6)  obtain  by  multi- 
plication the  trigonometric  fornnUas  for  sin((/)  —  0)  and  cos  (</>  —  B). 

17.  If  /,  ///.  )(  are  direction  cosines,  the  vect(ir  /i  +  ?/(j  +  )/k  is  a  vector  of  unit 
length  in  tlu'  direction  for  which  /,//(,  n  are  direction  cosines.  Show  that  (he 
condition  fm-  ])crpcndicularity  of  two  directions  (/,  m,  n)  and  (/'.  ;//'.  )(')  is 
IV  +  nmi  +  >"t'  =  0. 

18.  With  the  same  notations  as  in  Ex.  14  show  that 

j  i     j      k   I  Kyj   rf„  a.,  I 

a'LX  —  (!{  +  ((.J  +  (*.;-     anil      cixfi  =    a^   it„  a..  ;      ami      axft.y  —  1  h^    h„   //,,  i- 

i  \  ''■,  h  I  ; '■,  ''I  '•': : 


170 


DIFFER  EX  TI AL   ( '  ALC  U  LU  S 


19.  Compute  tlie  scalar  and  vector  products  of  these  pairs  of  vectors : 


(^'^)  i 


fv>i  +  0.3  j  —  5  k 


'[^0.1  i-  4.2]  +  2.0  k, 


(/^)  i 


r  i  +  2  j  +  3  k 


^  -  3  i  -  2  i  +  k, 


(7) 


ri  +  k 


20.  Find  tlie  areas  df  the  parallehigrams  dethied  Ijy  the  pairs  of  vectors  in 
Ex.  19.    Find  als(j  the  sine  and  cosine  of  the  angles  between  the  vectors. 

21.  Prove  ax[/3x(7x5)]  =  {a-yx5)(i  —  a-^yxS  =  /i-S  (xxy  —  j3-y  ax8. 

22.  What  is  the  area  of  the  triangle  (1,  1,  1),  (0.  2.  8).  (0.  0,-1)? 

77.  Vector  differentiation.  As  the  fundamental  rules  of  differentia- 
tion depend  on  the  laws  of  sul)traction.  inultiiilicatiou  by  a  nuinlx-r, 
the  distributive  law,  and  the  rules  pennittin.L;-  i-earrangement,  it  follows 
that  the  rules  must  l)e  ap^dicable  to  expi'essions  containing  vectors 
without  any  changes  except  those  implied  by  the  fact  that  axfS  =fc  /3xa. 
As  an  illustration  consider  the  application  of  the  definition  of  differen- 
tiation to  the  vector  jjroduct  Uxv  of  two  vectors  Avhic-h  are  supposed 
to  be  functions  of  a  numerical  variable,  say  ,/■.    Then 

A(Uxv)  =  (U  +  AUjx(V  +  AV)  —  UxV 
=  UxAv  -f  AUxV  +  AUxAv, 


A (UxV ) 
A.r 

r/(UxV) 


Av       AU 

Ux \ xV 

A,/'        A./' 


=   lim 


A (UxV ) 


AUxAv 

Jy      '/u 

ux     -  +  ■     -XV. 
(/./■         (t.r 


Here  the  ordinary  rule   for    a    product    is    seen  to    hold,  except  that 
///«  ordi'i'  (if  fltc  fdctoi'ii   III  list  nut  III'   Infci'cliinujiil . 

Th(^  interpretation  oi  tlie  derivative  is  inqioi-tant.  Let  the  variable 
vector  r  1)6  regarded  as  a  function  of  some  varial»le,  say  .'■,  and  suppose 
r  is  laid  off  from  an  assumed  origin  so  that,  as  ./■  varies, 
the  terminal  ])oint  of  r  describes  a  ctu've.  The  incre- 
ment Ar  of  r  coi-responding  to  A./'  is  a  vector  quantitv 
and   in    fact   is    the  cliord   of  the   ciu'\-c    as   indicatiMl. 

Tin-  (Irrirutn-r 


'/T       ,.      Ar 

— -  =  Inn    -  • 
>/.r  A./- 


..     Ar 

Inn    — 
A.s 


(42) 


is  ill  I' rr  fun-  II  rri-tni-  In  ivjriit  fn  flir  rii  riw  :  ill  ])articular  if 
the  variable  ./■  were  the  ai'c  x.  tlie  derivative  woidd  liavc 
tlie  magnitude  unity  and  would  be  a  unit  \'ector  tangent  to  the  curv(\ 
The  derivati\-c  or  differential  of  a  \ector  of  constant  length  is  per- 
pendicular  to   the    vector.     This    follows    from    the    lad    that   the   ^■cctor 


COMPLEX  XU3IBEK.S  AND  VECTORS 


171 


then  describes  a  circle  concentric  with  tlie  orij,dn.  It  may  also  be  seen 
analytically  from  the  e<j^uation 

r/(r.r)  =  ./r.r  +  r.^/r  =  2  r.r/r  =  d  const.  =  0.  (43) 

If  the  vector  of  constant  lenyth  is  of  length  unity,  the  increment  Ar  is 
the  chord  in  a  unit  circle  and,  apart  from  infinitesimals  of  higher 
order,  it  is  equal  in  magnitude  to  the  angle  subtended  at  the  center. 
Consider  then  the  derivative  of  the  unit  tangent  t  to  a  curve  with 
respect  to  the  arc  .s-.  The  magnitude  of  dt  is  the  angle  the  tangent  turns 
through  and  the  direction  of  dt  is  normal  to  t  and  hence  to  the  curve. 
The  vector  cpiantity,  i.        j-2~ 

cur\-ature     C  =  —  =  -7-^  >  (44) 

ds       (IS' 

therefore  has  the  magnitude  of  the  curvature  (by  the  definition  in  §  42) 
and  the  direction  of  the  interior  normal  to  the  curve. 

This  work  holds  equally  for  plane  or  space  curves.  In  tlie  case  of  a  space  curve 
the  plane  which  contains  the  tanirent  t  and  the  curvature  C  is  called  tlie  osculating 
Xjlane  (§  41).  By  definition  (§  42)  the  torsion  of  n  space  curve  is  the  rate  of  turning 
of  tlie  osculating  plane  with  tlie  arc.  that  is.  d-^/fU.  To  fin<l  the  torsion  by  vector 
methods  let  c  l>e  a  unit  vector  C/ v  C«C  along  C.  Then  as  t  and  c  are  perpendicular, 
n  =  txc  is  a  unit  vector  perpendicular  to  the  osculating  plane  and  rfn  will  ec^ual  d\p 
in  magnitude.  Hence  as  a  vect<ir  (piantity  the  torsion  is 
rZn      J  (txc) 


'7t  ,    '7c       .    (Zc 

—  xC  +  tx  —  =  tx —  , 
da  (/.s  dn  d-s  ds 


(45) 


where  (since  dt/ds  =  C.  and  c  is  parallel  to  C)  the  first  term 
drops  out.  Next  note  that  dn  is  iierpendicular  to  n  because  it 
is  the  differential  of  a  unit  vector,  and  is  perpendicular  to  t 

becatise  cZn  = 'Z(txc)  =  tx(?c  and  t«(txi?c)  =  0  since  t.  t.  dc  ;ivc       -^     _ 

necessarily   coniplaiiar   (Kx.  12.   p.  Ki'.i).     Hence  T   is  parallel  /  '     ^ 

to  c.    It  is  convenient  to  consider  the  torsion  as  iiositivc  \\iien        jj  / 
the  osculating  plane  seems  to  turn  in  tlic  positive  direction  when 
viewed  from  the  side  of  the  normal  plane  upon  whicli  t  lies.    An  inspection  of  tlie 
figure  shows  that  in  this  case  '7n  has  the  direction  —  c  and  not  +  c.    As  c  is  a  unit 
vector,  the  numerical  value  of  the  torsion  is  therefore  —  c-T.    Then 

'/       C 


T=  -cT 


=  —  C'tx 


.    dc 

C.tx-  = 
(Z.s 


ct. 


''■^- vCC 


(Ft      1 


+  C =z^ 

'l^-  ^''C.C         'l'^  VC-C 


■Ct) 


d-x      1 


ds'^  VCC 


(450 


^    C     d'^x 

t-  X 

C-C  ds^ 


X".l" 


where  differentiation  with  respect  to  .s  is  denoted  by  accents. 

78.  Another  sort  of  relation  between  vectors  and  differentiation 
comes  to  light  in  connection  with  the  normal  and  directional  deriva- 
tives (§  48).    If  /'(./■,  y,  z)  is  a  function  which  has  a  definite  value  at 


172 


DIFFEIIENTIAL   OALCULUS 


eiich  point  of  space  and  if  the  two  ncij^hboring  surfaces  F=  C  and 
F  =  ('  +  </('  are  considered,  tlie  normal  derivative  of  F  is  tlie  rate  of 
change  of  F  along  the  noi'nial  to  tlie  surfaces  and 
is  written  dF/<Jn.  The  rate  of  change  of  F  along  '  -  ^^^  ^ 
the  normal  to  the  surface  /•' =  ('  is  more  i-apid  than 
along  any  other  direction  ;  for  tlie  change  in  F  be- 
tween the  two  surfaces  is  (/F  —  di '  and  is  constant, 
whereas  the  distance  dn,  l)ctween  the  two  sui'faces  is 
least  (apai't  from  inhnitc'simals  of  higher  order)  along  the  normal.  In 
fact  if  dr  denote  the  distance  along  any  other  direction,  the  relations 
shown  by  the  figure  are 

dr  =  sec  Bdn     and     ——  =  ——  cos  6.  (46) 

(//•  (In 


If  now  n  denote  a  vectoi-  of  unit  length  normal  to  the  surface,  the 
product  XidF I dn  irlll  he  a,  iwctor  (jiKnitlti/  irliicli  Ihis  both  tlui  iiKKjnltitde 
and  tlie  direction  of  most  rapid  Increase  of  /•'.    Let 

dF 


dn 


iJ'ad  F 


(47) 


l)e  the  symbolic  expressions  for  this  vector,  wIkm'c  V/''  is  read  as  "del  /■'" 
and  grad  /•'  is  read  as  "  the  gi-adient  of  /•'."  If  dv  be  the  vector  of  which 
(//•  is  the  length,  the  scalar  product  n^/r  is  pre(dsely  cos  Odr,  and  hence 
it  follows  that 

,lx.\F=dF     and     r,.V/'  =  ^j  (48) 


where  r^  is  a,  unit  Ncclor  in  the  direction  dr.  The  second  of  tlie  ecpia- 
tioiis  shows  that  the  dlrcctlounl  dcrlcailrc  hi  a mj  direction  Is  tlie  com- 
ponent or  projection  of  tlie  (jrndloit  In  tliat  direction. 

From  this  fact  the  ex|)ression  of  the  gradient  may  be  found  in  terms 
of  its  components  along  the  axes.  Vur  the  derivatives  of  F  along  the 
axes  are  dF/cx,  cF Icij,  cF/cz,  and  as  these  are  the  comjjonents  of  \F 
along  the  directions  i,  j,  k,  the  result  is 


\l- 


HeiKu; 


^■rad  /•' 
V  =  i 


r.r 


c.z 


(49) 


c.r       ■'  ('1/  f:: 


may  be  regarded  as  a   symbolic  vcctor-dilferentiating  o])erator  Avhich 
when  applied  to  /•'  gi\'es  the  gradient  of  /•'.    The  ]»roduct 


dX'^I' 


dec 

'/,/■  ^-  -h  d,i  7  -  +  dr:   -  ]  I' 

c.r  •    cij  c::. 


dF 


m 


COMPLEX  NUMBERS   AND   VECTORS  173 

is  iiniiUHliately  seen  to  give  the  ordiiuiry  ex})ression  for  d]-\  l""roiu  this 
form  of  grad  F  it  does  not  a|)})ear  tliat  tlie  gradient  of  a  function  is 
independent  of  the  ehoice  of  axes,  but  fi'oju  the  manner  of  derivation 
of  VF  first  given  it  does  appear  that  grad  /•'  is  a  (h'finite  vectoi-  (}uan- 
tity  independent  of  the  ehoiee  of  axes. 

In  the  ease  of  any  given  function  /-'  the  gradient  may  he  found  l)y 
the  application  of  the  formuhi  (-^9);  hut  in  many  instances  it  may  also 
he  found  hy  means  of  the  important  relation  dr'\ F  =  dF  of  (-1-8).  For 
instance  to  prove  the  formula  \  <^F(<)  =  FVG  -\-  G'Vi-',  the  relation  may 
1)0  applied  as  follows  : 

(/r.V(/-V/)  =  d(Ff;)  =  F<Ur  4-  (idF 

=  FdX'Xa  +  (;dX'\F ^  dr.{FV(;  +  ^tf). 

Now  as  these  equations  hold  for  any  dircH'tion  dr,  tlu"  dx  may  he  can- 
celed by  (35),  p.  IGo,  and  the  desired  result  is  obtained. 

The  list'  of  vector  notations  for  treating;-  assiu-ncd  practical  prohlenis  involving 
computation  is  not  i^reat,  but  for  haudliiii;'  tlic  i^ciicral  tiicory  of  such  parts  of 
physics  as  are  essentially  concerned  with  direct  iiuantities,  mechanics,  hydro- 
mechanics, electromagnetic  theories,  et('.,  tlie  actual  use  of  the  \'ector  a(u'ovisnis 
considerably  shortens  the  formulas  and  has  the  added  advantau'e  of  operating- 
directly  upon  the  magnitudes  involved.  At  this  point  some  of  the  elements  of 
mechanics  will  be  developed. 

79.  According  to  Xc\vton"s  Second  Law,  when  a  force  acts  upon  ;i 
])article  of  mass  m.  tlie  i'<iti>  of  clKiniji'  of  iiuiiiu'ntii in  is  ei/xcl  to  tlic 
fori'c  (icinKj.  (Did  tiil:i's  jdiicr  in  tin'  dirrctinn  of  flic  force.  ]t  therefore^ 
a})})cars  that  the  I'ate  of  clumge  of  momentum  and  momentum  itself 
are  to  1h'  regarded  as  vector  or  directed  magnitudes  in  the  a])|)licati()n 
of  the  St'cond  Law.  Now  if  the  vector  r,  laid  olT  I'roni  a  fixed  origin 
to  the  point  at  which  the  moving  mass  in  is  situated  at  tiny  insttmt  of 
time  t,  l)e  differentiated  with  respect  to  the  time  t,  tin'  derivative  dx jdt 
is  a  vector,  tangent  to  the  ciu've  in  which  the  particle  is  moving  and  of 
mttgnitude  e(]ual  to  ds/dt  or  /•.  the  velocity  of  motion.  As  vectors*, 
then,  th(>  velocity  v  ttnd  the  momentum  ;ind  the  ft)rce  nitiy  be  written  ;is 


Hence 


dx 
V  ==  -— ' 

dt 

my, 

F  =  'rjniv). 
lit 

dv          d-x 

III    ■,-    =  III  -—; 
dt              dt' 

=  inf 

(ol) 


From  the  equation.s  it  ap])ears  thttt  the  force  F  is  tlie  ])ro(luct  of  the 
mass  ///  by  a  vector  f  which  is  the  rate  of  change  of  the  velocity  rcganknl 

*  In  ai)i)licatioiis,  it  is  usual  to  denote  vectors  by  heavy  type  and  to  denote  the  magni- 
tudes (if  tliose  vectors  by  corresponding  italic  letters. 


174  DIFFERENTIAL  CALCULUS 

as  a  vector.  The  vector  f  is  called  the  acriderdtinn :  it  must  not  l>e  con- 
fused with  the  rate  of  change  dr/df  or  d'-s/i/f-  of  the  speed  or  magnitude 
of  the  velocity.  The  components  f^,  f^,  J\  of  the  acceleration  along  the 
axes  are  the  projections  of  f  along  the  directions  i,  j,  k  and  may  be 
■written  as  f  •!,  f  •]',  f 'k.    Then  by  the  laws  of  differentiation  it  follows 

that  ,  ,  /     -x        7 

.  _  r  .  _  '^V    .  _  d  (y.l)  _  f/r,, 

'^  ~    '^~  dt''^~      dt      ~  dt  ' 

-  .        d-r  .        d-{T'i)        d-.r 

or  tr  =  I •!  =  -TT.'!  =  TT, —  =  -TT,  • 

dt-  dt-  dt- 

d'.r  d'lf  (Pz 

Hence  /,,  =  — ;  ^         /„  =  ~,^         /'^  =  17 ' 

(It'  dt'  dt 

and  it  is  seen  that  the  com])onents  of  the  acceleration  are  the  acceler- 
ations of  the  com})onents.  If  A',  )',  Z  are  the  components  of  the  force, 
the  equations  of  motion  in  re(;tangular  coordinates  are 


in 


d'.r  (T'li  d'z 

—7  =  A',  m  -y';  =    y,  W  -^7.  =  Z.  (o2) 

dt-  '  dt-  '  dt-  ^     ' 


Instead  of  resolving  the  acceleration,  force,  and  displacement  along 
the  axes,  it  may  be  convenient  to  I'csolve  them  along  the  tangent  and 
normal  to  the  curve.  The  velocity  v  may  be  written  as  /-t,  wliere  /•  is 
the  magnitude  of  the  velocity  and  t  is  a  unit  vector  tangent  to  the 
curve.    Then  ,  ,    ^ 

r/v  _  d(r\.)   _  dr  ilX. 

~Yt  ~       dt       ~  "dt      ^  ''Vt 
dt        <lt  ds        ^  r 

But  -  =  ,,^:  ;77  =  C-  =  ^  n,  (o3) 

where  E  is  the  radius  of  curvature  and  n  is  a  unit  normal.    Hence 
,       d-s  r-  d's  r- 

It  therefore  is  seen  that  the  component  of  the  acceleration  along  t!ie 
tangent  is  d-s/dt'\  or  the  rate  of  change  of  tlie  velocity  i-egarded  as  a 
numl)er,  and  the  (•onn.)on(Mit  noi'iiial  to  the  curxc  is  r-/Jl.  If  7'  and  A' 
are  the  comjtonents  of  the  force  along  the  tangent  and  normal  to  the 
curve  of  motion,  the  e(piations  are 

'^'•^'  .  '•' 

"/'  ^  I"  ft  =  III    r".,  '  A"  ^  iiif,^  =  11)  — 

dt-  '  It 

It  is  noteworthy  that  the  force  must  lie  in  the  osculating  ])lane. 

If  r  and  r  -f-  Ar  are  two  positidiis  of  tlie  I'adius  \-i'ctor.  tlie  area  of 
the  seetoi-  included  bv  them  is  i  I'Xcept  \in-  infinitesimals  of  higlier  (jrder) 


COMPLEX  XUMBEES  AND   VECTORS  175 

AA  =  i  rx(r  +  Ar)  =  i  rxAr,  and  is  a  vector  quantity  of  whieli  the 
direction  is  normal  to  tlie  plane  of  r  and  r  +  Ar,  that  is,  to  the  plane 
through  the  origin  tangent  to  the  curve.  The  rate  of  description  of  area, 
or  the  ureal  vdoc'dii,  is  therefore 

— -  =  Inii  I  rx —  =  ,  rx— -  =  \  rxv.  (54) 

dt  -      \f       -      dt       -  ^     ' 

The  in-ojections  of  the  areal  velocities  on  the  coordinate  planes,  which 
are  the  saiiu'  as  the  areal  velocities  of  the  projection  of  the  motion  on 
those  planes,  art-  (Ex.  11  l)elo\v) 

dn  d,i^  1/    d.r  dz\  1  /    d,i  d.r\      _,, 

^^^''^)'         2['^~'''d-f)'         2\'"di~^^dj)-   ^^^) 

If  the  force  F  acting  on  the  mass  m  passes  through  the  origin,  then 
r  and  F  lie  along  the  same  direction  and  rxF  —  0.  The  equation  of 
motion  may  then  be  integrated  at  sight. 

m  — -  =  F,  7//rx— -  =  rxF  =  0, 

dt         '  df  ' 

'^v       r/  ^       ,       ^ 
rx— -  —  —  (rxv)  =  0,         rxv  =  const. 
dt       dt^       ■'         ' 

It  is  seen  that  in  this  case  the  rate  of  description  of  area  is  a  constant 
vector,  Avhich  means  that  the  rate  is  not  only  constant  in  magnitude 
but  is  constant  in  direction,  that  is,  the  path  of  the  particle  m.  must  lie 
in  a  plane  through  the  origin.  When  the  force  passes  through  a  fixed 
point,  as  in  this  case,  the  foice  is  said  to  l)e  ccnfrdl.  Therefore  when  a 
particle  moves  under  the  action  of  a  central  force,  the  motion  takes  place 
in  a  plane  passing  through  the  center  and  the  rate  of  description  of 
areas,  or  the  areal  velocity,  is  constant. 

80.  If  there  are  several  i)articles.  say  n.  in  motion,  eacli  has  its  own  equation 
of  ni(jtion.  Tliese  e(iuatiuiis  may  t»e  eomljined  by  adilition  ami  .sul)sequent  reduction. 

m,        '  =  F,.  »;.,  —  -  =  F.,.  •  •  •.   m„  — "  =  F„, 
^  dt-  '        -  dt-  -  dt- 

d-T,  d-r„  d-T„      ^        ^  ^ 

and  »ii    ,  '  +  ?Ho  -^o"  +  •  •  •  +  '""  -T^  =  F,  +  F.  +  ■  •  •  +  F„. 

^  J-r,  d-T.-,  d-x„       d-  , 

But         m,  — -  +  m„  — -  +  •  ■  •  +  »;„ =  —  (m,r,  +  "'or,,  +  ■  ■  •  +  mnT„). 

1  fifi  -  at'-  dt-       df^  ^    ^  ^         -  - 

Let  m^x^  +  m.-x.^  +  •  •  •  +  m,j„  =  {jn^  +  m.-,  +  ■  •  •  +  )n„)  f  =  3/  f 

-  _  '"I'^i  +  '"■>''•>  +  ■  ■  •  +  "hJn  __  2»(r  _  2»ir 


or 


?/(j  +  »(.,  +  •••  +  "In  Swi  M 


Then  .V-^  =  F,  +  F.,  +  •  •  •  +  F„  =  Vf.  (55) 

dt-         1         -  ^     "       ^ 


170  ])IFFEKENT1AL   CALCULUS 

Now  the  vector  r  whicli  has  been  here  iutnKluced  is  tlie  vector  of  tlie  center  of 
mass  or  center  of  gravity  of  the  particles  {Ex.  .'>,  p.  1()8).  'I'lie  result  {')o)  states,  on 
comparison  with  (51),  that  the  centc'r  of  gravity  of  the  n  masses  moves  as  if  all  the 
mass  M  were  concentrated  at  it  and  all  the  forces  applied  thi-re. 

The  force  F;  acting  on  the  itli  mass  may  be  whdlly  or  partly  due.  to  attractions, 
I'cpulsions,  pressures,  or  other  actions  exerted  on  that  mass  by  one  or  more  of  the 
other  masses  of  the  system  of  n  particles.    In  fact  let  F,-  be  written  as 

F,-  =  F,,,  +  F/i  +  F,-^  +  •  •  •  +  F„,, 

where  F,-/  is  the  force  exerteil  on  »/,■  Ijy  ;//,  and  F,-,,  is  llie  force  due  tn  some  agency 
external  to  I  he  ii  masses  which  f(n'ni  the  system.  Now  l)y  Newton's  Tliird  Law, 
when  one  particle  acts  upon  a  second,  the  secoml  reacts  upon  the  tirst  with  a 
force  which  is  e(|ual  in  magnitude  and  opposite  in  direction.  Hence  (o  F/,  above 
there  will  correspond  a  force  F/,-  =—  F/y  exerted  by  im  on  mj.  In  the  sum  i;F,-  aU 
these  equal  and  opposite  actions  and  reactions  will  drop  out  and  >;F/  may  be  ir~ 
l)laced  by  SF,-o,  the  sum  of  the  external  forces.  Hence  I  Ik.' iiiiiiortant  theorem  that  : 
The  inotion  of  the  renter  of  indss  of  a  .sci  of  'particles  is  (ts  if  (ill  the  muss  ii'crc  co)icv)i- 
iratcd  there  (oul  all  the  external  forces  vjere  apj)lied  there  (the  infernal  fcn-ces,  that  is, 
the  forces  of  nuitual  action  and  reaction  between  the  particles  being  entirely 
neglected). 

The  moment  of  a  force  about  a  given  point  is  defined  as  the  pi-oduct  of  tlie  force 
by  till'  perpt'udicular  distance  of  the  force  from  the  ])oiut..  If  r  is  the  vector  from 
the  jxiint  its  origin  to  any  point  in  the  line  of  the  force,  the  moment  is  therefori; 
rxF  when  considered  as  a  vectcu'  iiuantity,  and  is  perpendicular  to  the  i)lane  of  tin; 
line  of  the  force  and  the  (u-igin.  The  e(iuations  of  »  mo\ing  masses  may  now  be 
combinetl  in  a  different  way  and  reduced,  ^lultiply  the  eipiations  by  r^,  r.,,  •  •  •,  r,j 
and  add.    Then 

'^v,  '?v.,  f?v„  ^  .„  „ 

7H,r,x       '  4-  )H„r„x      -  4-  .  .  .  -(-  //(,,r„x       -  =r,xF,  +  r.,xF.,  +  •  •  •  +  r„xF,, 

d  d  d  ^  ^  ^ 

or      m,      r.xv,  +  /«.,      r.,xv.,  +  ■  •  •  +  ni„    -  r„xv„  =.  r.xF,  +  r.,xF„  +  •  •  •  +  r,,xF,, 

d 
or  -  (»(,r|XV^  +  ///„r.,xv.,  +  .  .  .  -f  //(„r„xv„)  :=  iilrxF.  (50) 

'I'his  ('((nation  shows  tliat  if  the  a  real  velocities  of  the  dift'ereiit  masses  are  multiplied 
by  those  masses,  and  all  added  together,  the  derivative  of  the  sum  obtained  is  e(|ual 
to  the  nioinent  of  all  the  forces  about  the  origin,  the  moments  of  the  different  forces 
being  added  as  \-ector  (luantities. 

This  i-esult  may  be  simiilitied  and  put  in  a  different  form.  Consider  again  the 
resolution  of  F,  into  the  sum  F/u  +  F;i  +  •  •  •  +  F,„.  and  in  ]iarticular  cousiilei-  the 
action  F,;/  and  the  reaction  F;,  -:  —  F;/  belwi'eii  two  iiarticles.  Let  it  be  assumed 
that  the  action  and  reaction  are  not  only  ('<]ual  and  opposite,  but  lie  along  the  line 
coniiectiiig  the  two  particles.  Then  tlie  i)er])eiidicular  distances  from  the  oi'iiiin  to 
the  actioii  and  reaction  are  ('(Uial  and  the  moments  of  the  action  and  reaction  ai'e 
■'(lual  and  opposite,  ;uid  ma\'  be  dropjx'd  from  the  sum  2r;xF,-.  which  then  reduces 
to  i;r,xF,ii.  (  Ml  the  other  hand  a  term  like  iii;r;:<v,-  may  be  written  as  r,x(/;(,v/).  This 
jiroduct  is  foi'ined  from  the  nionientum  in  exactly  the  saiiu!  way  that  the  niomenl 
is  fornied  from  tlie  force,  and  it  is  called  the  moment  of  momentum.  Hence  the 
e(iuation  (5(i)  becomes 


COMPLEX  LUMBERS  AND   ^'ECT(:)KS  177 

—  (total   inoineiit  of   inoinentiuii)  =  moment  of  external  forces. 
dt  ^  ' 

Hence  the  result  that,  as  vector  (juantities  :  T/ie  r<Ue  of  change  of  the  moment  of 
momentum  of  a  system  of  particles  is  equal  to  the  moment  of  the  external  forces  (the 
forces  between  the  masses  being  entirely  nef,'-lected  under  the  assumption  that  action 
and  reaction  lie  along  the  line  connecting  the  masses). 

EXERCISES 

1.  Apply  the  deJiiiition  of  differentiation  to  prove 

(a)  d(U'V)  =  U'dv  +  V'du,         (/3)  (/[u.(vxw)]  =  rfu.(vxw)  +  u.('Zvxw)  +  u.(vx'/w). 

2.  Differentiate  under  tlie  assumption  that  vectors  denoted  l)y  early  letttM's  of 
the  alphabet  are  constant  and  tliose  designated  by  the  later  letters  are  variable  : 

(a)  ux(vxw),         (p)  acos^  +  bsin/,         (7)   (u-u)  u, 

(5)  ux       ,  (e)  u.    ,    X     -    ,  (j-)  c(a.u  . 

dx  \ilx    dx-l 

3.  Apply  the  rules  for  change  of  variable  tr>sho\v  that  -  -  =  — '- '- — ,  where 

ds-  s'-'' 

accents  denote  differentiation  with  respect  to  x.  In  case  r.  =  ^1+  //j  show  that 
1/ vC»C  takes  th(>  usual  form  for  the  radius  of  curvatiii-i'  of  a  plane  curve. 

4.  The  e(]uation  of  the  helix  is  r  =  i'l"  cos  <p  -\-  ]a  sin  0  +  kJxp  with  .s  =  a  ar  +  6-  0  ; 
show  that  the  radius  of  curvature  is  («-  +  h-)/a. 

5.  Find  the  torsion  of  the  helix.    It  is  b/{a-  +  0-). 

6.  Change  the  variable  from  .s  to  some  otiier  variable  /  in  the  fornnila  for  torsion. 

7.  In  the  following  cases  find  the  gradient  either  by  ai)plying  the  fornnila  which 
contains  the  partial  derivatives,  or  by  using  llie  relation  dT-VF  —  dF,  or  both  : 

(a)  r.r  =  ./•-  +  //-  +  z-.         Hi)  log,-,         (7)  /•  =  Ar^. 
(5)  log  (.r- +//-)  =  log  [r.r  -  (k.r)-],        (t)   (rxa).(rxb). 

8.  Prove  these  laws  of  operation  with  the'  symbol  V  : 

(a)  V(F  +  G)  =  VF  +  Vr;,  (p)  G-\(F/G)  =  CYF -  F\(i. 

9.  If  r.  (p  are  polar  corirdinates  in  a  plane  ami  r,  is  a  tiiiii  vector  along  the  radius 
vector,  show  thrd  i\r^/dt  =:  nd<p/dt  where  n  is  a.  tuiit  \-eeliu-  peri)endieular  to  tlie 
radius.  Thus  differentiate  r  = /Tj  twice  and  separate  the  result  iiUo  components 
along  the  radius  vector  and  perpendicular  to  it  so  that 

'~"dt:^         \rf(/  '  *"'"(Z>       "  dt  dt"  rdt\     dt, 

10.  Prove  conversely  to  the  text  that  if  the  vector  rate  of  description  of  area  is 
constant,  the  force  nuist  be  central,  that  is.  rxF  =  0. 

11.  Note  that  rxy.i.  rxyj.  rxv-k  are  the  xn'ii.if'Ctions  of  the  areal  velocities  upon 
tlie  plaiu's  x  =  0,  //  =  0,  z  ~  0.    Hence  derive  (-34')  of  the  text. 


178  DIFFEEEXTIAL  CALCULUS 

12.  Show  that  the  Cartesian  exi^ressions  for  the  inafrnitude  of  the  velocity  and 
of  the  acceleration  and  for  the  rate  of  chanii:e  of  the  speed  dv/dt  are 

V  =  Vx'^  +  v'-  +  z'-,    /  -  Vx"-  +  y"-  +  z"-,     t'  = — ^^^—   -    -  , 

Vx'-  +  U"-  +  z'- 

where  accents  denote  differentiation  with  respect  to  the  time. 

13.  Snppose  that  a  bodj^  winch  is  riirid  is  rotating  about  an  axis  with  the 
anjiular  velocity  w  =  dcp/dt.  Represent  the  an^-ular  velocity  by  a  vectcn-  a  drawn 
along  the  axis  and  of  magnitude  equal  to  w.  Show  that  the  velocity  of  any  poini 
in  space  is  v  =  axr.  where  r  is  the  vector  drawn  to  that  point  from  any  point  (jf 
the  axis  as  origin.  Show  that  the  acceleration  of  the  ix)int  determined  l)y  r  is  in  a 
plane  through  the  point  and  perpendicular  to  the  axis,  and  that  the  components  are 

ax(axr)  =  (a»r)a  —  w'-r  toward  the  axis,      {da./dt)y.x  perpendicular  to  the  axis, 

under  the  assumption  that  the  axis  of  rotation  is  invariable. 

14.  Let  f  denote  the  center  of  gravity  of  a  sy.stem  of  particles  and  r/  denote  the 
vector  drawn  from  the  center  of  gravity  to  the  ith  particle  so  that  r,  =  f  +  r/  and 
V,  =  V  +  v/.    The  kinetic  energy  of  the  /'th  particle  is  by  definition 

\  nuvf  =  I  »i,V{.v,-  -  .V  ;»;(v  +  v/).(v  +  v/). 

Sum  up  for  all  particles  and  simplify  by  using  the  fact  Smj-r^  =  0,  which  is  due  to 
the  assumption  that  the  origin  ftu'  the  vectors  T-  is  at  the  center  of  gravity.  Hence 
prove  the  important  theorem  :  The  total  kinetic  energy  of  a  system  is  t'(jiuil  to  the 
kinetic  energy  which  the  total  mass  v-ould  have  If  moving  unth  the  center  of  gravity 
plus  the  energy  computed  from  the  motion  relative  to  the  center  of  gravity  as  origin, 
that  is, 

r  =  i  Zmirf  =  i  Mr-  +  \  Sm^rf. 

15.  Consider  a  rigid  body  moving  in  a  plane,  which  may  be  taken  as  tlie  x//- 
plane.  Let  any  point  r,-,  of  the  body  ).)e  marked  and  other  points  be  denoted  rela- 
tive to  it  by  r'.  The  motion  of  any  point  r'  is  compounded  from  the  motion  of  r,^ 
and  from  the  angular  velocity  a  =  kw  of  the  body  alioiu  the  jioiiu  r,,.  In  tact  the 
velocity  v  of  any  ])oint  is  v  =  v,,  +  axr'.  Show  tliat  the  velocity  of  the  ])oiiit  di-notccl 
by  r' =  kxv,|/co  is  zero.  Thisjioint  is  known  as  the  instaiUaneons  cenrer  of  rotation 
(§  3!)).  Show  that  the  corirdinat(.'s  of  the  instantaneous  center  referred  to  axes  at 
the  origin  of  the  vectors  r  are 

1  d//,,  .  1  (/x„ 

X  =  r.i  =  x,j -" ,         //  ==  r.j  =  //„  +    -       ' . 

w  (It  oj  dt 

16.  If  several  forces  Fj.  F..  •  ■  ■.  F„  act  on  a  l)ody.  tlie  sum  R  =  ZF,  is  called 
the  resultant  and  the  siun  Zr,-xF,-.  wlieix'  r,-  is  drawn  from  an  m-igin  0  to  a  point 
in  the  line  of  the  f(n'<'e  F,-.  is  called  the  resultant  momoit  about  O.  Show  that  the 
resultaiU  moments  Mo  and  Mo'  about  two  })oints  are  connected  by  the  relation 
Mo'  =  Mo  -i-  Mo'(Ro).  when'  Mo'(Ro)  nieaiis  the  moment  about  O'  of  the  resultant, 
R  considereil  as  applied  at  O.  Infer  that  moments  about  all  points  of  any  line 
parallel  to  tlie  resultant  are  eijual.  Show  that  in  any  plane  perpendicular  to  R 
thert'  is  a  ])oiiU  ()'  L:iven  by  r  =  RxMo/R-R.  where  0  is  any  point  of  tlie  plane, 
such  that  Mo'  is  parallel  to  R. 


PART  II.    DIFFERENTIAL  EQUATIONS 

CHAPTER   VII 

GENERAL   INTRODUCTION    TO   DIFFERENTIAL   EQUATIONS 

81.  Some  geometric  problems.  The  application  of  the  differential 
caleulus  to  plane  curves  has  given  a  means  of  determining  some 
geometric^  properties  of  the  curves.  For  instance,  the  length  of  the 
suLnormal  of  a  curve  (§  7)  is  ij<]t//(l.f,  which  in  the  case  of  the  parabola 
//"  =  -ipv  is  2/(,  that  is,  the  subnormal  is  constant.  Suppose  now  it 
were  desired  conversely  to  find  all  curves  for  which  the  subnormal  is 
a  given  constant  7;^  The  statement  of  this  problem  is  evidently  con- 
tained in  the  equation 

'^!f  f  7  7 

?/  —  =  m-     or     2/1/'  =  m.     or     ydy  =  viax. 

Again,  the  radius  of  curvature  of  the  lemniscate  /•-  =  <r  cos  2  <^  is  found 
to  1)6  11  =  a-^/'dr,  that  is,  the  radius  of  curvature  varies  inversely  as  the 
ludius.  If  conversely  it  were  desired  to  find  all  curves  for  whi(;h  the 
radius  of  curvature  varies  inversely  as  the  radius  of  the  curve,  the  state- 
ment of  the  probh^m  would  be  the  equation 


'  +  ^% 


h 


,1-r  ^  /  dr 

where  /•  is  a  constant  called  a  factor  of  ])r()|)ortionality.* 

Equations  like  these  are  unlike  ordinary  algebraic  ecjuations  l)ecause, 
in  addition  to  the  varial)les  x,  1/  or  /•,  cf>  and  certain  constants  m  or  /,-, 
they  contain  also  derivatives,  as  dy/dr  or  (h-/d(fi  and  d'r/i/<f)'~,  of  one  of 
the  variables  Avitli  res])ect  to  the  othei'.    An  equation  which  contains 

*  ^SlaiiY  prdhlcms  in  uronii'try,  iiiecliaiiics,  aod  physics  are  stated  in  terms  of  varia- 
tion. For  i>urposes  of  analysis  tlie  statement  x  varies  as  //.  or  x  x  if,  is  written  as  x  --  kij, 
introdneini;  a  I'onstant  k  ealled  a  factor  of  proportionality  to  convert  the  variation  into 
an  equation.  In  like  manner  the  statement  .'■  varies  inversely  as  //,  or  x  -r.  \/\i,  becomes 
X  =-  k/ii,  and  ./•  vai'ies  jointly  witli  //  and  z  becomes  .'•  —  kiiz. 

1711 


180  DIFFEREXTIAL   EQUATIONS 

derivatives  is  called  a  d'ijfcrenfhij  riiiKitlnn.  The  orili'r  of  the  differential 
equation  is  the  order  of  the  highest  derivative  it  contains.  The  equa- 
tions above  are  respectively  of  the  first  and  second  orders.  A  differen- 
tial equation  of  the  first  order  may  be  symbolized  as  $  (./■,  ij,  ij)  =  0, 
and  one  of  the  second  order  as  4>(,/',  v/,  ij\  y")  =  0.  A  function  i/  =/(./•) 
given  explicitly  or  defined  implicitly  by  the  relation  F{x,  //)  =  0  is 
said  to  be  a  solntlan  of  a  given  differential  equation  if  the  e<iuation  is 
true  for  all  values  of  the  independent  variable  ./•  Avhen  the  ex})ressions 
for  1/  and  its  derivatives  are  substituted  in  the  e(|uation. 

Tlius  to  show  that  {wn  matter  what  the  vahie  of  a  is)  the  rehation 
4  (ly  —  J-  +  2  a-  hit;-  x  =  0 
gives  a  sokition  of  the  differential  e<iuatif>n  of  the  second  order 

Xdx)  \dx-/ 

it  is  merely  necessary  to  form  the  derivatives 

(1;/  ft-  il-y  a- 

•2  a  —  —  X ,         -In  —  =  1-1 

dx  X  dx-  X- 

and  substitute  tliem  in  the  given  equation  together  with  //  to  see  that 

'  +  („;■)  - '- fc)  = '  +  4,7=  {■'- -  ^ "^  +  .:=)  -  4 .-^ ('  +  :?r  +  ii)  =  " 

is  clearly  satisfied  for  all  values  (if  ./■.  It  appears  therefore  that  the  given  relation 
fur  ij  is  a  solutiiiu  of  the  given  equation. 

To  infi'firiifi'  or  snlrr  a  differential  eipiation  is  to  find  all  the  fuiietioiis 
which  satisfy  the  equatinii.  (Jeonictrically  s})eaking.  it  is  to  find  all  the 
curves  whicli  have  the  property  expressed  by  the  e(]uation.  In  median- 
ics  it  is  to  find  all  possible  motions  arising  from  the  given  forces.  The 
method  of  integrating  or  solving  a  differential  e(piation  depends  largely 
upon  the  inf/m/fif//  of  the  solver.  In  many  cases,  however,  some  method 
is  immediatelv  obvious.  l''or  instance  if  it  be  jtossible  to  sfjiuraff  fhr 
i-iirlitlili's,  so  that  the  diffenMitial  '///  is  mulri])lied  by  a  function  (jf  // 
alone  and  tlx  by  a  fiun-tion  of  ./■  alone,  as  in  tlie  cquati(jn 

cfii '/)</'/  — ij/ (.'■}(/.!',      then        /  c/)  I//) '///=/(//(.'•)'/.'•+  *"  (1) 

will  clearly  be  the  integral  or  solution  of  the  ditferential  equation. 

As  an  exainjile.  let  the  eurves  of  eoustant  subnormal  be  deterinineil.    Here 

1,'di/  =-  iiidx     ami      ;/-  r-  2 //(./•  -f-  ('. 

The  variables  are  already  separutetl  and  the  integration  is  inmieiliate.  The  eurves 
are  parabolas  with  semi-latus   rectum   equal   to  the  constant   and   with  the  axis 


GENERAL  INTRODUCTION  181 

coincident  witli  the  axis  of  x.  If  in  particular  it  were  desired  to  deterniiue  tliat 
curve  whose  subnormal  was  m  and  wliirli  passed  throuirh  the  oriiiin,  it  would 
merely  be  necessary  to  substitute  (0,  0)  in  the  ecjuation  y-  =  2mx  +  C  to  ascertain 
what  particidar  value  nuist  be  assiuned  to  C  in  order  that  the  curve  pass  through 
(0,  0).    The  value  is  C  =  0. 

Another  example  miiiht  be  to  determine  the  curves  for  which  the  x-intercept 
varies  as  the  abscissa  of  the  point  of  taniiency.  As  the  expression  (§  7)  for  the 
x-intercept  is  £  —  ijdjc/dij.  the  statement  is 

dx  /I       ,\  <^-^ 

s  —  II  —  =  kz     or     (\  —  k)x  =  y  — 
dy  dy 

Hence  (1  —  A)    --  ^   —     and     (1  —  A)  lo--  q  =  loux  +  G. 

'J        -t 

If  desired,  this  expression  may  be  ciiaiii^ed  to  another  form  by  using  each  side  of 
the  e(iuality  as  an  exponent  with  tiie  base  e.    Then 

(.(l-i)Iog)/  =  (Xo^x  +  C      iji-      ij\-k  =  t^'x  =  C'x. 

As  G'is  an  arljitrary  constant,  the  constant  C"  =  e'-'is  also  arbitrary  and  the  solution 
may  simply  lie  written  as  y^~'^  =  Cx.  where  the  accent  has  been  onntted  from  the 
constant.  If  it  were  desired  to  pick  out  that  particular  curve  which  passed  through 
the  i)oint  (1,  1),  it  would  merely  be  necessary  to  determine  C  from  the  e(iuation 

li-'.-  ==  CI,     and  hence     C  ^  1. 

As  a  third  example  let  the  curves  whose  tangent  is  constant  and  equal  to  «  be 
determined.    The  length  of  the  tangent  is  // vi  +  ;/'-///'  and  hence  the  equation  is 


Vi  +  //'-  .,  1  +  //'-  \'^:-  —  //-   , 

y ■=  (I     or     y- —    =a     or     1  =  y 

u'  u"-  U 

The  variables  are  therefore  separable  and  the  results  are 


\'((--y-                                         '..-..          ,      ii  +  V(i-—y- 
dx  = '('/     and     x  +  (    =  \  ii-  —  i/-  —  <t  Ut's 

y        '  '  ti 

If  it  be  desired  that  the  tangent  at  the  origin  be  vertic'al  so  that  the  curve  passes 
through  (0.  «).  tiie  constant  C  is  0.  The  curve  is  the  tractrix  or  "curve  of  pursuit"' 
as  described  by  a  calf  dragged  at  the  entl  of  a  rope  by  a  person  walking  along 
a  straight  line. 

82.  Pr(»l)lfins  wliieli  involve  tlic  radius  of  cui'vaturc  will  lead  to  differ- 
ential ('([uations  of  the  si-cond  (H-dei-.  The  nierhod  of  solving'  such 
])rol)lenis  is  to  n-ilitci'  fjir  cijinifijui,  if  iinssilih',  to  unc  <if  flw  p rst  Di'ilcr. 
For  the  se'cond  derivatiN'e  may  he  written  as 

and  /;  =  t^+^^  =  li±4^  =  1±jft  (r, 

U  ''11  , ''// 

dx  dij 


182  DIFFERENTIAL  EQUATIONS 

is  the  expression  for  the  radius  of  curvature.  If  it  be  given  that  tlie 
radius  of  curvature  is  of  the  form /'(./■)(/)  (y')  ov  /(//)  eft  {;/'), 

^pi^f,,,^,y,    „v    (l+£l=fy,^<fy         (3) 

the  variables  .r  and  //'  or  y  and  ij'  are  immediately  separable,  and  an 
integration  may  be  performed.  This  will  lead  to  an  efj^uation  of  the 
first  order ;  and  if  the  variables  are  again  se})aruble,  the  solution  may 
be  completed  by  tlie  methods  of  the  alxjve  cxam})les. 

In  the  first  place  consider  curves  wliose  radius  of  curvature  is  constant.    Tlien 

(1  +  w"-)^                           da'            dx          ,            //             /  —  C 
^        -^    ■^    =  a     or =  —     and  '  = , 

d/  (1  +  r-)'^        «  Vl  +  l/'-  « 

dx 

where  the  constant  of  integration  has  been  written  as  —  C/a  for  future  conven- 
ience. The  equation  may  now  be  solved  for  ;/'  and  the  variables  become  separated 
with  the  results 

■r  -  C  ,  (.r  -  C)  , 

(J    =   =^^=1^=       or       dl/  =    ;^z::^^3^z=:z=r  dX . 

V«-  -  {x  —  cy-  Vk-  —  (.(•  —  cy- 


Hence         y  —  C"  =—  Va-  —  {x  —  Cy-     or     (x  —  (')'-  +  (//  —  Cy-  =  a'-. 

The  curves,  as  shonld  be  anticipated,  are  circles  of  radius  a  and  with  any  arbi- 
trary point  (C,  C")  as  center.  It  should  be  nnted  that,  as  the  solution  lias  recjuired 
two  successive  integ-rations,  there  are  two  arbitrary  constants  C  and  C"  of  integra- 
tion in  the  result. 

As  a  second  example  consider  the  curves  wlmsc  radius  of  curvature  is  dunble 
the  normal.    As  the  length  of  the  normal  is  y  V  1  -f  //"-,  the  equation  becomes 


i — i-  =  2i/Vl+  I/-     or     — - 

, '('/  ,  dl/ 

y  -^~  >i  ~^- 

dy  dy 


i  -i-  y. 


where  the  dotible  sign  has  been  introduced  when  the  radical  is  removed  by  cancel- 
lation. This  is  necessary  ;  for  before  the  cancellation  the  signs  were  ambii:uinis 
and  there  is  no  reason  to  assume  that  the  ambiguity  disappears.  In  fact,  if  the 
curve  is  concave  up,  the  second  derivative  is  positive  and  the  I'adius  nf  c-ur\aturf 
is  reckoned  as  p(»sitive,  whereas  tlie  normal  is  positive  or  negative  accnrdinu"  as 
the  curve  is  above  or  below  the  axis  nf  x  :  similarly,  if  the  curve  is  concave  down. 
Let  the  negative  sign  be  chosen.  This  corresponds  to  a  curve  aliove  the  axis  and 
concave  down  i>i  lielow  the  axis  and  concave  up,  that  is,  the  normal  and  the  radius 
of  curvature  have  the  same  direction.    Then 

—  =  -  ~-^'-^     and     log  y  =  -  log  (1  +  //'-)  +  log  2  C. 

y         1  +  .'/'- 

where  the  constant  has  been  uiven  the  form  log  2  ('  for  ron\-enience.  This  expres- 
sion may  be  thrown  into  alurliraie  form  by  exponentiation.  sol\ed  for  //'.  and  then 


GENEKAL   IXTKODUCTIOX  183 


y{l-\-  y'-)  =  2G     or     y-  = or  z  =  dx. 

y  V2  Cy  -  y' 


Hence  x  —  6"  =  C  vers- ^  — ;  —  Vz  (Jy  —  y'^. 

The  curves  are  cycloids  of  which  tlie  generating  circle  has  an  arbitrary  radius  (' 
and  of  which  the  cusps  are  upon  the  j-axis  at  the  points  C"  i  2  kirC.  If  the  posi- 
tive sign  had  been  taken  in  the  eciuation,  the  curves  would  liave  been  entirely 
different  ;  see  Ex.  5  (a). 

The  number  of  arljitrary  eoiistaiits  of  integration  Avhicli  enter  into 
the  solution  of  a  differential  equation  depends  on  tlie  number  of  inte- 
grations which  are  ])erformed  and  is  equal  to  the  order  of  the  equation. 
This  results  in  giving  a  family  of  curves,  dependent  on  one  or  moi-e 
parameters,  as  tlie  solution  of  tlie  equation.  To  pick  out  any  particular 
member  of  the  family,  additional  conditions  must  be  given.  Thus,  if 
there  is  only  one  constant  of  integration,  the  curve  may  be  requiri'd 
to  pass  through  a  given  point;  if  there  are  two  constants,  the  curve 
may  be  required  to  pass  through  a  given  point  and  liave  a  given  slope 
at  that  point,  or  to  jiass  through  two  given  i)oints.  These  additional 
conditions  are  called  Inltbd  condltloiis.  In  meclianics  the  initial  condi- 
tions are  very  important ;  for  the  point  reached  Ijy  a  particle  describing 
a  curve  under  the  action  of  assigned  forces  depends  not  only  on  the 
forces,  but  on  the  point  at  which  the  ])article  starttnl  and  the  velocity 
with  which  it  started.  In  all  cases  the  distinction  between  tlie  consijtnta 
of  lute  (//'Of  ton  and  tlie  (jlnni  consfdufs  of  fin'  jn'ohlcm  (in  the  foregoing 
examples,  the  distinction  between  (',  C  and  //>,  /.■,  ")  should  Ije  ke])t 
clearly  in  mind 

EXERCISES 

1.  Verify  the  solutions  of  the  differential  equations  : 

(a)  xy  +  1,  x^  ^C,   //  -h  J-  +  xy'  =  0,  (/i)  .r'//-  (:]  f'-+  C)  =  1,  ,r//'+  //-h.rV^f-'-=  0. 

(7)   {\\xr-)y'-^\^  2.f=r;f"-r'-ie^.",     (5)   y  -F  ,/•//'  =  xhf-.,  xy  =  C-x  +  C. 

(e)  y"  +  ^'A  =  0,  y  =  c  lo-./-  +  C\,     (f)  //  =  rt'-  +  C'lC- '■.  y"  +  2  //  =  8  //'. 
,  ^    ///  )         ,.         _i,./^,        .f  A  o      ,,    .   .fA';j\ 

iv)  y     —  y  =  •<-,  //  =  <~  t-'-  +  e    2  ■  I  Cj  cos  — ^  +  <  o  sm  — ^  I  —  x-. 

2.  Determine  the  curves  which  have  tlie  following  properties: 

{a)  The  subtangent  is  constant  ;  y'"-  =  Ce^.    If  Ihrougli  (2,  2).  //'"  =  2">e^-". 

(/3)  The  right  triangle  furnied  by  the  tangent,  subtangent,  and  ordinate  has  the 
constant  area  k/2  ;  the  liyperbolas  xy  -f  Cy  +  k  =  0.  SIkiw  that  if  the  curve  ;[)asscs 
through  (1,  2)  ami  (2.  1).  tlie  arbitrary  constant  C  is  0  and  the  given  /,:  is  —  2. 

(7)  The  normal  is  constant  in  length  ;  the  circles  (x  —  C)-  +  y-  =  k-. 

(5)  The  normal  varies  as  the  sijuare  of  the  ordinate  ;  catenaries  t//  =  cosli  k{x—  C). 
If  in  particular  the  curve  is  perpendicular  to  the  (/-axis.  (/  —  0. 

(f )  The  area  of  the  right  triangle  furnied  by  the  tangent,  normal,  and  .c-axis  is 
inversely  proportional  to  the  slope  ;  the  circles  (./•  —  (')'-  +  y-  =  /.■. 


184  DIFFERENTIAL  EQUATIONS 

3.  Dc'terinine  the  curves  whicli  have  the  following  properties: 

(a)  The  angle  between  the  radius  vector  and  tangent  is  constant ;  spirals 
r  =  Ce'^'t'. 

(/3)  The  angle  between  the  radius  vector  and  tangent  is  half  that  between  the 
radius  and  initial  line  ;  cardioids  /•  =  6'(1  —  cos0). 

(7)  The  perpendicular  from  the  pole  to  a  tangent  is  constant ;  r  cos  (^  —  ( ')  =  k. 

(5)  The  tangent  is  eipially  inclined  to  the  radius  vector  and  to  the  initial  line  ; 
the  two  sets  of  parabolas  r  =  C'/(l  ±  cos  0). 

(c)  The  radius  is  eijually  inclined  to  the  normal  and  to  the  initial  line  ;  circles 
r  =  C  cos  (p  or  Hues  r  cos  cp  =  C. 

4.  The  arc  s  of  a  curve  is  proportional  to  the  area  A,  where  in  rectangular 
coordinates  A  is  the  area  under  the  curve  and  in  polar  coordinates  it  is  the  area 
included  by  the  curve  and  the  radius  vectors.  From  the  e(iuation  (Z.s  =  (Z.l  show 
that  the  curves  which  satisfy  the  condition  are  catenaries  for  rectangular  coordi- 
nates and  lines  for  polar  coordinates. 

5.  Determine  the  curves  for  which  the  radius  of  curvature 

((1-)  is  twice  the  normal  and  oppositely  directed  ;  parabolas  (x  —  C)"  =  C"{2 y  —  C). 

(/3)  is  eiiual  tn  the  normal  and  in  same  din'ction  ;  circles  (x  —  Cy~  +  y-  =  C"-. 

(7)  is  ecpial  to  tiie  normal  and  in  opposite  direction  ;  catenaries. 

(5 )  varies  as  the  cube  of  the  normal  ;  conies  kCy'^  —  C'^  {x,  +  G")'-^  =  A". 

( e )  projected  on  the  x-axis  equals  the  abscissa  ;  catenaries. 

( f )  projected  on  the  x-axis  is  the  negative  of  the  abscissa  ;  circles. 
{■>])  projected  on  the  x-axis  is  twice  the  abscissa. 

(6)  is  proportional  to  the  slope  of  tin;  tangent  or  of  the  normal. 

83.  Problems  in  mechanics  and  physics.  In  many  ])liysif'al  prol)lems 
the  stateininit  involves  an  e(ina,tion  between  the  I'tifi'  of  rlKiagc  of  some 
quantity  and  the  value  of  that  (juantity.  In  this  way  the  solution  of 
the  problem  is  made  to  depend  on  the  integration  of  a  differential  e(pia- 
tion  of  the  first  order.  If  x  denotes  any  quantity,  the  rate  of  in(;reas(» 
in  .r  is  (I.r/(lf  and  the  rate  of  decrease  in  .r  is  —  dx/dt ;  and  consequently 
when  the  rate  of  change  of  x  is  a  function  of  r/',  the  variables  are 
immediately  sepai'ated  and  the  integration  may  be  ])ei'formed.  The 
constant  of  integration  has  to  be  determined  fi'om  the  initial  conditions  ; 
till!  constants  inlierent  in  tlie  ])r()lilem  nuiy  be  given  in  advam-e  or  their 
values  may  be  detenuined  by  comparing  ,'■  and  t  at  some  sid)se<|uent 
tune.  The  exei'cises  offei-ed  below  Avill  exemplify  the  ti'eatnu'iit  of 
sucli  ])ro])lems. 

In  other  ])liysieal  problems  tlu',  statement  of  the  question  as  a  differ- 
ential ('(piation  is  not  so  direct  and  is  carried  out  by  an  examination  of 
the  problem  witli  a  view  to  stating  a  I'elation  Itetween  the  increnu'uts 
or  diffei'entials  of  the  de])endent  and  independent  vai'iables,  as  in  sonu; 
geomcti'ie  I'elations  already  discussed  (^i  40).  and  in  tlie  ])rol)lem  of  tlu; 
tension  in  a  ro[te  wra]>])i'd  around  a  cxlindrieal  post  discussed  below. 


GENERAL   IXTKODUCTIOX 


185 


T+Ar 


Y 


pA'A.s 


The  method  may  be  further  illustrated  by  tlie  derivation  of  the  differ- 
ential equations  of  the  curve  of  e<|uilibriuui  of  a  flL'xil)le  string  or 
chain.  Let  p  be  the  density  of  the  chain  so  that  p\s  is  the  mass  of 
the  length  Ax;  let  A'  and  Y  be  the  components 
of  the  force  (estimated  per  unit  mass)  acting  on 
the  elements  of  the  chain.  Let  T  denote  the 
tension  in  the  chain,  and  r  the  inclination  of 
the  clement  of  chain.  From  the  figure  it  then 
a})i»cars  that  the  components  of  all  the  forces 
acting  on  A.s-  are 

(7'  -f  AV;  COS  (r  +  Ar)  —  T  cos  r  +  ApAx  =  0, 
(T  +  A  7')  sin  (t  +  Ar)  -  T  sin  r  +  }>A.s-  =  0 ; 

for  these  must  be  zero  if  the  element  is  to  be  in  a  position  of  equi- 
librium.   The  eipiations  may  l)e  written  in  the  form 

A  (  7'  cos  r )  +  A  p  A.s  =  0,  A  (  T  sin  r)  +  3  >  Ax  =  0  : 

and  if  they  now   be  divided  by  Ax  and  if   Ax  be  allowed  to  approach 
zero,  the  result  is  the  two  c(piations  of  e(prilibrium 


X 


(b 


d 


where  cos  r  and  sin  r  are  re})laced  by  tlieir  values  (Li'./</s  and  (Jt//<7s. 


-„'^// 


If  tlie  string  is  acted  ciu  mily  hj  furres  parallel  to  a  uivcii  ilircctidii.  let  the 
//-axis  be  taken  as  parallel  to  that  direction.  'I'lien  the  component  -V  will  be  zero 
and  the  lirst  equation  may  be  integrated.    '1"1k'  result  is 


T 


dx 


(' 


r=  c 


d^ 
dx 


This  value  of  T  may  be  substituted  in  the  second  equation.   There  i.s  thus  obtained 
a  differential  ecpiation  of  the  second  order 


.|')..r  =  o 


or     C 


Vl  -1-  (/'- 


+  py 


0. 


(4') 


r-i-A.7' 


If  this  equation  can  be  integrated,  tlie  form  of  the  curve 
of  equilibrium  may  be  found. 

Another  problem  of  a  different  nature  in  strings  is  to 
ci  insider  the  variation  of  the  tension  in  a  rope  wound  around 
a  cylinder  without  overlapping.  The  forces  acting  on  the 
element  Ax  of  the  rope  are  the  tensions  T  and  T  +  AT,  the 
normal  pressure  or  reaction  T!  of  the  cylinder,  and  the  force 
of  friction  whicli  is  proportional  to  the  pressure.  It  will 
be  assumed  that  the  normal  reaction  lies  in  the  angle  A(f>  and  that  the  coelhcient 
of  friction  is  fx  so  that  the  force  of  friction  is  /xU.  The  components  along  the  radiu.s 
and  along  the  tangent  are 


186  DIFFERENTIAL    EQrATlOXS 

{T  +  AT)  sin  A4>  -  R  cos  (^A^)  -  /jlR  sin  (ftA-p)  =  0,         0  <  ^  <  1, 
(r  +  AT)  COS  A0  +  R  sin  [OA'P)  -  /xR  cos  ((9A0)  -  T  =  0. 

Now  discard  all  infinitesimals  except  those  of  the  first  order.  It  must  be  borne  in 
mind  that  the  pressure  R  is  the  reaction  on  the  infinitesimal  arc  As-  and  hence  is 
itself  infinitesimal.  The  substitutions  are  therefore  Tdcp  for  {T  +  AT)  sin  A0,  R  fur 
R  cos  OA(p.  0  fur  R  sin  dAcp.  and  7'  +  dT  for  (2'  +  A 7')  cos  A(p.  The  equations  there- 
fore reduce  to  twu  sinqile  e(iuations 

Ta<p-R  =  0.         dT-iJLR  =  0, 

from  which  the  unknown  A'  may  be  eliminated  with  the  result 

dT  — /xTdcp     or      T  =  Ce>^'i>     or     T  —  T^,e>^<t>, 

where  7'^  is  the  tension  when  <p  is  U.  The  tension  therefore  runs  up  exponentially 
and  affords  ample  explanation  of  why  a  man.  by  winding  a  rope  about  a  post,  can 
reailily  hold  a  sliip  or  otlier  object  exertinu-  a  great  force  at  the  other  end  of  the 
i'o;)c.  If  /J.  is  1/3.  three  turns  about  the  post  will  hold  a  force  5:>-3  T^^,  or  tiver  25 
tons,  if  the  man  exerts  a  force  of  a  hundredweight. 

84.  If  a  constant  mass  di  is  moving  along  a  ]in(^  under  the  influence 
of  a  force  F  acting  along  the  line,  Xewton's  Set-ond  Law  of  Motion  (p.  13) 
states  the  j^roUem  of  the  motion  as  the  differential  ecjuation 

■mf  =  F     or     III  ~r~,  =  F  (o) 

of  the  second  order  ;  and  it  therefore  appears  tliat  the  comiJlete  solution 
of  a  ])rol)lem  in  rectilinear  motion  requires  the  integration  of  this  eijua- 
tion.    The  acceleration  may  Ije  written  as 

■'  ~  dt''  d.r  df  ~  '\77 ' 
and  hence  the  equation  of  motion  takes  either  of  the  forms 

F.  (5') 

It  now'  a|)pears  that  thei-e  are  several  cases  in  which  the  tirst  integration 
may  l)e  performed.  l-^)r  if  the  f(jrce  is  a  functi(jn  of  the  velocity  or  of 
the  nine  or  a  ])roducr  of  two  such  functions,  the  varial>les  are  separated 
in  the  tirst  form  (jf  the  e(juation  :  whereas  if  the  force  is  a  function  of 
tin;  velocity  or  of  the  coordinati'  .'•  or  a  jiroduct  of  two  such  functions, 
the  variables  ai'C  sepai'ated  in  the  second  foi'm  of  the  eipiation. 

When  the  tii'st  integration  is  jiei-formed  according  to  either  of  these 
methods,  tliei-i-  will  arise  an  equation  l)etween  the  vehjcity  and  either 
the  tinu^  f  or  the  c(.)ordinate  .'•.  In  this  equation  ^\■ill  he  contained  a 
constant  of  integration  wliich  may  he  detei-miiu^l  hy  the  initial  condi- 
tions, tliat  is.  l)v  the  kiio\\le(hjv  of  the  velocitv  at  tlie  start,  whetlier  in 


,lr 

dr 

,77  =  ^ 

or 

iiir 

dr 

GEXEKAL   INTKODUCTKJX  187 

time  or  in  position.  Finally  it  will  be  possiljle  (at  least  theoretically) 
to  solve  the  equation  and  express  the  velocity  as  a  function  of  the  time 
t  or  of  the  position  ./■,  as  the  case  may  be,  and  inte.qrate  a  second  time. 
The  cariying  through  in  ])ractice  of  this  sketcli  of  the  Avork  will  be 
exemplihed  in  the  following  two  examples. 

Suppose  a  particle  of  mass  m  is  projected  vertically  upward  with  the  velocity  V. 
Solve  the  problem  of  the  motion  under  the  assumption  that  the  resistance  of  the 
air  varies  as  the  velocity  of  tlie  particle.  Let  the  distance  be  measured  vertically 
upward.  The  forces  actin:,^  on  the  particle  are  two,  —  the  force  of  uravity  which  is 
the  weiirht  11'=  mfj.  and  tlie  resistance  of  the  air  which  is  A'r.  IJoth  these  forces 
are  lu-frative  because  they  are  directed  toward  diminishing  values  of  s..    Hence 

ml  =:  —  iii(i  —  kv     or     »)-_  =  _  (nr/  _  A-f, 
di 

where  the  first  form  of  the  eqttation  of  motion  has  been  chosen,  although  in  this 
case  the  second  form  would  be  equally  available.    Then  integrate. 

-  -  dt     and    log  ( ^  +  —  i- )  = t  +  C. 


k 
g  +  -  V 

m 


As  by  the  initial  conditions  v  =  V  when  t  =  0.  tiie  constant  C  is  found  from 

k  _ 

,      /         k  ,A  k  ^      ^,       ,  ''  '^  m^         -^,' 

log  (^  H 1  )  =  —  -    0  +  f  ;     hence 


g  +  -V 
in 

is  the  relation  between  i;  and  /  found  by  substituting  the  value  of  C.  The  solution 
for  V  gives 

dx       /))i,  ^A    — '       m 

'  =  dt  =  {:k'-'')' '"  --k''- 

"'  /"I        ,  A  — '      »'    , 

Hence  x  = -ff  +  1     '■    '"   -      7^'  +  (' ■ 

k\k  I  k 

li  the  particle  starts  from  the  origin  j-  =  0.  the  constaiu  C  is  found  to  be 

Hence  the  positinn  nf  the  particle  is  expressed  in  terms  of  tiie  tiim.'  and  tlie  prob- 
lem is  solved.  If  it  be  desired  to  lind  the  time  which  elapses  before  the  particle 
collies  to  rest  and  starts  to  drop  back,  it  is  merely  necessary  to  siilistitute  r  =  0  in 
the  relation  connectinir  tlie  velocity  ami  the  time,  and  solve  for  the  time  t  z=  T : 
and  if  this  value  of  t  be  substituted  in  the  expression  for  x,  the  total  distance  A' 
covered  in  the  ascent  will  be  found.    Tlie  results  are 


4  --(fr[^'-'-^-i^')j- 


k         \        rug 


As  a  second  example  consiiler  the  mr>tion  of  a  jiarticle  vilirating  up  and  down 
at  the  end  of  an  elastic  strinu-  held  in  tlie  liehl  of  L:ravitv.    I'v  Hooke's  Law  for 


188  DIFFERENTIAL   EQUATIONS 

t'lastic  strings  tlie  force  exerted  by  the  string  is  proportional  to  the  extension  of 
the  string  over  its  natural  length,  that  is,  F  =  kAl.  Let  /  be  the  length  of  the  string, 
AJ  the  extension  of  the  string  ju,st  sut!ieient  to  hold  the  weight  11'=  mg  at  rest  so 
that  frAy/ =  my,  and  let  x  measured  downward  be  the  additional  exti^nsion  of  the 
string  at  any  instant  of  the  motion.  The  force  of  gravity  mg  is  positive  and  the 
force  of  elasticity  —  /i'(A|/  +  x)  is  negative.  The  second  form  of  the  equation  of 
motion  is  to  be  chosen.    Hence 

my  —  —  mg  —  A'  {AJ  +  i)     or     mv  —  =  —  Lr,     snice     mg  —  kAJ. 

(xJu  tl%C 

Then  mi-di-  =  —  kj:dx     or     mv-  =  —  kf~  +  (' . 

Suppose  that  x  =  a  is  the  amplitude  of  the  motion,  so  that  when  ,f  =  ii  the  velocity 
u  =  0  and  the  particle  stops  and  starts  back.    Then  C  =  A''(-.    Hence 

(Is  k     ,— ,  as  ~k   , 

V  =  —  =  \       \  (I-  —  s'~     or     — —  -  --  =  A       (u. 


and 


.sin-i  -  z=  \  -^t  +  C     or    x  ~  n  sin  \  \\-  t  -{■  0' )  • 
a        \  m.  \M  m  / 


Now  let  the  time  be  measured  from  the  instant  when  the  particle  passes  through 
the  position  s  =  0.  Then  C  satisties  the  e(juation  0  =  n  sin  ('  and  may  be  taken  as 
zero.  The  motion  is  theref(n'e  given  by  the  equation  s  =  a  >-ui  \  k/mt  and  is 
periodic.  While  t  changes  by  2  tt  \^m/k  the  particle  completes  an  entire  oscilla- 
tion. The  time  T  =  27r  \'m/k  is  called  the  2)cr iodic  time.  The  motion  considered 
in  this  example  is  characterized  by  the  fact  that  the  total  force  —  Z,-/  is  propor- 
tional to  the  displacement  from  a  certiiiii  oi-ii^in  and  is  directed  toward  the  origin. 
Motion  of  this  sort  is  called  simple  lnuinmiie  motion  (brietij-  S.  II.  M.)  and  is  of 
great  importance  in  mechanics  and  physics. 

EXERCISES 

1.  The  sum  of  SI  00  is  put  at  interest  at  4  per  cent  per  ainium  under  the  condition 
that  the  interest  shall  be  comjjounded  at  each  instant.  Show  that  the  sum  will 
amount  to  .Sl'OO  in  17  yr.  4  mo.,  and  to  ,^1000  in  57|  yr. 

2.  Given  that  the  rate  of  decomposition  of  an  amount  s  of  a  given  substance  is 
proportional  to  the  amount  of  the  substance  remaining  undccomposcd.  Solve  tlie 
problem  of  the  decom])nsition  and  determine  the  constani  of  integration  and  the 
physical  constant  of  proportionality  if  s  =  o.ll  when  /  =  (J  and  ./•  =  1.4!S  wiu-ii 
(  =  40  mill.     Ahs.  k  -  .OoOO. 

3.  -V  substance  is  luidergoing  transformation  into  another  at  a  rate  wliii'h  is 
assumed  to  be  ])ropoi-ti(inal  to  the  aniouiu  of  the  substance  st  ill  remaining  untratis- 
formed.  If  that  amount  is  o.'j.Ci  when  t  =  1  hr.  and  I^.S  wlien  /  =  4  lir..  iletermine 
the  amotuit  at  the  start  when  /  =  0  and  the  constant  of  proportionality  and  lind 
how  many  hours  will  elapse  before  only  one-thousandth  of  the  original  ammuit 
will  remain. 

4.  If  the  aetivity  A  of  a  radioactive  deposit  is  ]n-oportional  to  its  rate  of 
diminution  and  is  found  to  decrease  to  ',  its  initial  value  in  4  days,  show  that  ^1 
.satisties  the  eiiuation  -!/.!„  =  (,-"•'"■''. 


GENERAL  IXTKODUCTIOX  189 

5.  Suppose  tliat  amounts  a  and  b  respectiveh'  of  two  substances  are  involved  in 
a  reaction  in  wliicli  tlie  velocity  of  transformation  dx/dt  is  proportional  to  the  prod- 
uct (a  —  x){b  —  x)  of  the  amounts  remaining  untransformed.  Integrate  on  the 
supposition  that  a  ^t  b. 

t     \  a  —  X  \  b  —  X 
log  ^'^Li)  =  („  _  },)  kt  ;     and  if     lluij" ,  0.48(j(j '  0.2342 
"  ^''  "  -^^  120.3 1 0.3870  \  0. 1354 

determine  the  product  k{a  —  b). 

6.  Integrate  the  ecjuation  of  Ex.  5  if  a  —  b.  and  determine  a  and  k  it  x  =  0.87 
when  t  —  15  and  x  =  13.00  when  t  =  55. 

7.  If  the  velocit}^  of  a  chemical  reaction  in  which  three  substances  are  involved 
is  proportional  to  the  continued  product  of  the  amounts  of  the  substances  remaining, 
show  that  the  equation  between  x  and  the  time  is 

,  a  -  % 


"■"  \a  -  x)       [b  -  x)       [c  -  x) 


(x  =  0 
—  —  kt,     where     -{ 


{<(  -  b)(b-  c)(c-  a)  '  [_t  =0. 

8.  Solve  Ex.  7  if  a  =  b  7^  c  ;  also  when  a  =  b  =  c.  Note  the  very  different 
forms  of  the  solution  in  the  three  cases. 

9.  The  rate  at  whicli  water  runs  out  of  a  Vduk  tlirough  a  small  pipe  issuing 
horizontall}'  near  the  bottom  of  the  tank  is  proportional  to  the  s(iuare  root  of  the 
height  of  the  surface  of  the  water  above  the  pipe.  If  the  tank  is  cylindrical  and 
half  empties  in  30  niiii..  show  that  it  will  completely  empty  in  about  100  min. 

10.  Discuss  Ex.  0  in  case  the  tank  were  a  right  cone  or  frustum  of  a  cone. 

11.  Consider  a  vertical  colunui  of  air  and  assiune  that  the  pressure  at  any  level 
is  due  to  the  weight  of  the  air  above.  Show  that  p  =ji^fi-^'''  gives  the  jiressure  at 
any  height  h,  if  Boyle's  Law  tl:at  the  density  of  a  gas  varies  as  the  pressure  be  used. 

12.  Work  Ex.  n  under  the  assumption  that  the  adiabatic  law  pxp'-'*  rejtre- 
seiUs  the  conditions  in  the  atmosphere.  Show  that  in  this  case  the  pressure  would 
becf)me  zero  at  a  Unite  height.  (If  the  proper  numerical  data  are  inserted,  the 
height  turns  out  to  be  about  20  miles.  The  adiabatic  law  seems  to  correspond 
better  to  the  facts  than  Boyle's  Law.) 

13.  Let  I  be  the  natural  length  of  an  elastic  .string,  let  Al  be  the  extension,  and 
assume  Ilooke's  Law  that  the  force  is  proportional  to  the  extension  in  the  form 
A/  =  klF.  Let  the  string  be  held  in  a  vertical  position  so  as  to  elongate  under  its 
own  weight  IT.    Show  that  the  elongation  is  lk]Vl. 

14.  The  density  of  water  under  a  pressure  of  p  atmosi^heres  is  p  =  1  +  0.00004 p. 
Show  that  the  sitrface  of  an  ocean  six  miles  deep  is  aljout  000  ft.  below  the  position 
it  would  have  if  water  were  incompressilile. 

15.  Show  that  the  eiiuations  of  the  curve  of  e(itii]ibrium  of  a  string  or  chain  are 

in  polar  coordinates,  where  7i  and  ^  are  the  components  of  the  force  along  the 
radius  vector  and  perpendicular  to  it. 


190  UIFFEHKNTIAL    I^K'ATIOX.S 

16.  Slinw  that  (IT  +  pSih  =  0  and  7"+  pi:X  =  0  aro  tlu-  (M]uati(iMs  of  <M|uilib- 
riiini  of  a  .striu-i;'  if  /.'  is  tiic  radius  of  curvatun'  and  .S  and  .V  are  the  taiiL,^eiiLial  and 
normal  compononts  of  the  forces. 

17.*  Show  that  when  a  unifoi'ni  chain  is  supported  at  two  points  and  lianas  down 
betwi'en  the  points  under  its  own  weight,  the  curve  of  equilibrium  is  the  catenary. 

18.  Suppose  the  mass  ihii  of  the  element  rZ.s  of  a  chain  is  proportional  to  the  pro- 
jection (l.r  of  (Is  on  the  .f-axis,  and  that  tlie  chain  han,i;K  in  the  field  of  i;ravity. 
Show  that  the  cur\('  is  a  parabola.  (This  is  essentially  the  problem  of  the  shape 
of  the  cabk's  in  a  suspension  bridn'e  when  the  roadbed  is  of  unifoi'm  linear  density  ; 
for  tlu'  weii^ht  of  thi'  cables  is  ne^ligiljle  compared  to  that  of  the  roailbed.) 

19.  It  is  desired  to  sti'inii;  upon  a  cord  a.  i^reat  many  uniform  heavy  rods  of 
varying;  lenj,^ths  -so  that  when  the  cord  is  huns  up  with  the  rods  dauirling  from  it 
the  rods  will  be  eipially  spaced  along  the  horizontal  and  have  their  lower  ends  on 
the  same  level.  Required  the  shape  the  conl  will  take.  (It  should  be  noted  that 
the  shai)e  must  be  known  before  the  rods  can  be  out  in  tlu'  pnjper  lengths  t(j  hang 
as  desired.)    The  weight  of  the  cord  may  be  neglected. 

20.  A  masoiu'V  arch  carrit's  a  horizontal  roadbed.  On  the  assumption  that  the 
material  between  the  arch  and  the  roadbed  is  of  unifiu'm  density  aiid  that  eacli 
(lenient  of  the  arch  supports  the  weight  of  the  material  al)ove  it,  find  the  shape  of 
the  arch. 

21.  In  equations  (4')  the  integration  may  be  carried  through  in  terms  of  ijuadra- 
tures  if  pV  is  a  function  of  //  alone  :  and  similarly  in  Ex.  15  the  integration  may  be 
carried  through  if  <t)  =  0  and  pl\  is  a  function  of  r  alone  so  that  the  Held  is  central. 
Sliow  that  the  results  of  thus  carrying  through  the  integration  are  the  formulas 

G(///  r  ( 'dr/r 


x  + 


/('(hi  r  (  (Ir/r 

■V(fpYdy)-  -  C^  J    -^(fplUhf  -  C^ 


22.  .\  Y)article  falls  from  rest  through  the  air,  which  is  assumed  to  offer  a  resist- 
aut'c  ]iroportional  to  the  velocity.  Solve  the  problem  with  the  initial  conditions 
V  =  0.  .r  =  0.  t  =  0.  Sho.w  that  as  the  particle  falls,  the  velocity  does  not  increa.se 
indelinitely,  but  appi'oaches  a  definite  limit  T=  »';///i'. 

23.  Solve  I'^x.  22  with  the  initial  conditions  i' =  r,,.  .r  =  0.  /  =  0,  where  r,,  is 
gi'eater  than  the  limiting  velocity  ]'.   .Show  that  the  particle  slows  down  as  it  falls. 

24.  \  particle  rises  througli  the  air.  wliich  is  assumed  to  resist  proportionally  to 
the  s(iuare  of  the  velocity.    Solve  the  motinu. 

25.  Solve  the  problem  analogous  to  Kx.  24  for  a  falling  particle.  Show  tliat 
there  is  a  limiting  velocity  V  =  \  iiKj/k.  If  the  iiarticle  were  projected  down  with 
an  initial  velocity  greater  than  V.  it  would  slow  ilowu  as  in  Ex.  23. 

26.  A  ]iarl  icle  falls  towards  a  point  which  attracts  it  inversely  as  the  S(iuare  of  the 
distance  ami  dii'ectly  as  its  mass.  Find  the  relation  between  x  and  t  and  determine 
the  total  time  T  taken  to  reach  the  center.    Initial  conilitions  r  =  0.  .r  =  a.  t  =  0. 

\  I  =       (MIS  -        +  ^   (/,(■  —  .(-.  /    ==  TT/.- 

>  a  2  a 

*  l''\ercisrs  17~2U  slieuM  l>c  worked  <(l,  hiitin  hy  the  iiicthnd  liy  wliich  (4)  were  derived, 
not  by  ai)i)lyiiiu-  (  h  <lircctly. 


ge:neeal  i^Ti:()])r('Ti()x  191 

27.  A  particle  starts  from  tlie  orii^in  with  a  velocity  T"  and  moves  in  a  medium 

which  resists  proportionally  to  the  velocity.    Find  the  relations  between  velocity 

and  distance,  velocity  and  time,  and  distance  and  time  ;  also  the  limiting  distance 

traversed. 

-tt  -'If 

v=  V  —  kx/m,         V  =  Ve    '"  ,         kx  =  ?h  l'(l  —  e    '"  ),         mV/k. 

28.  Solve  Ex.  27  under  the  assumption  that  the  resistance  varies  as  vu. 

29.  A  particle  falls  toward  a  point  which  attracts  inversely  as  the  cube  of  the 
distance  and  directly  as  the  mass.  The  initial  conditi"ons  are  x  =  (/.  v  =  0.  t  =  0. 
Show  that  x-  =  a'-—  kt"/a-  and  the  total  time  of  descent  is  T  =  <i-/\  k. 

30.  A  cylindrical  spar  buoy  stands  vertically  in  the  water.  The  buoy  is  pressed 
down  a  little  and  released.  Show  that,  if  the  resistance  of  the  water  and  air  be 
neglected,  the  motion  is  simple  harmonic.  Integrate  and  iletermine  the  constants 
from  the  initial  conditions  x  =  0,  v  =  V.  t  =  0.  where  x  measures  the  displacement 
from  the  position  of  ecjuilibrium. 

31.  A  particle  slides  down  a  rough  inclined  plane.  Determine  the  motion.  Note 
that  of  the  force  of  gravity  only  the  component  ;/(f/ sin  t  acts  dnwn  the  plane, 
whereas  the  component  mij  cos  ;  acts  perpendii'ularly  to  the  plane  and  develops  the 
force  /uL))tg  cos  i  of  friction.  Here  i  is  the  inclination  of  the  plane  and  fx  is  the 
coetRcient  of  friction. 

32.  A  bead  is  free  to  move  upon  a  frictionless  wire  in  the  form  of  an  inverted 
cycloid  (vertex  down).  Show  that  the  component  of  the  weight  along  the  tangent 
to  the  cycloid  is  proportional  to  the  distance  of  the  particle  from  the  vertex.  Hence 
determine  the  motion  as  simple  harmonic  and  lix  the  constants  of  integration  by 
the  initial  conditions  that  the  particle  starts  from  rest  at  the  top  vt  the  cycloid. 

33.  Two  equal  weights  are  hanging  at  the  end  ()f  an  elastic  string.  One  drops 
off.    Determine  completely  the  motion  of  the  particle  remaining. 

34.  One  end  of  an  elastic  s])ring  (such  as  istised  in  a  spring  balance)  is  attached 
I'igidly  to  a  point  on  a  horizontal  table.  To  the  other  end  a  particle  is  attached. 
If  the  particle  be  held  at  such  a  point  that  the  spring  is  elongated  by  the  amount 
a  and  then  released,  determine  the  motion  on  the  assumption  that  the  coefficient 
of  friction  between  the  particle  and  the  table  is  fj. ;  and  discuss  the  possibility  of 
different  cases  according  as  the  force  of  friction  is  small  or  large  relative  to  the 
force  exerted  by  the  spring. 

85.  Lineal  element  and  differential  equation.  The  idea  of  a  curve 
as  juade  \i\)  of  the  points  upon  it  is  familial-.  Points,  however,  have  no 
extension  and  therefore  must  Ije  regarded  not  as  pieces  of  a  curve  but 
merely  as  positions  on  it.  Strictly  speaking,  the  pieces  of  a  curve  are 
the  elements  A.s-  of  arc;  hut  for  many  purposes  it  is  convenient  to  vi'- 
place  the  com})licated  element  A.s  by  a  piece  of  the  tangent  to  the  curve 
at  some  point  of  the  arc  A.s-,  and  from  this  point  of  view  a  curve  is  made 
U])  of  an  infinite  number  of  infinitesimal  elements  tangent  to  it.  This 
is  analogous  to  the  point  of  view  by  which  a  curve  is  regarded  as  made 


192  DIFFEKENTIAL   EQUATIONS 

up  of  an  intiiiite  number  of  infinitesimal  cliords  and  is  intimately  related 
to  the  conception  of  the  curve  as  the  envelope  of  its  tangents  (§  65). 
A  point  on  a  curve  taken  witli  an  infinitesimal  ])ortion  of  the  tangent 
to  the  curve  at  tliat  })oint  is  called  a  Uncal  element  of  tlie  curve.  These 
concepts  and  definitions  are  clearly  ecjually  available  in  two  or  three 
dimensions.  For  the  present  the  curves  under  dis- 
cussion will  be  plane  curves  and  the  lineal  elements 
will  therefore  all  lie  in  a  plane.  ''^  f(xyn) 

To  specify  any  particular  lineal  element  iln-ee 
eoordbvitcs  :r,  i/,  p  will  be  used,  of  Avliich  tlie  two  (./•,  //)  determine  the 
point  through  which  the  element  passes  and  of  which  the  third  p  is 
the  slope  of  the  element.  If  a  curve  /'(./',  //)  =  0  is  given,  the  slope  at 
any  point  may  be  found  by  differentiation, 

p  =  y-  =  —  —  /  —  J  (6) 

(/,/■  C.I-/    c//  ^  ^ 

and  hence  the  third  coordinate  p  of  the  lineal  elements  of  this  particular 
curve  is  expressed  in  terms  of  the  other  two.  If  in  }ilace  of  one  curve 
f(.r,  >/)  =  ()  the  whole  family  of  curves  /(■'',  >/)  =  <",  Avhere  C  is  an 
arbitrary  constant,  had  been  given,  the  slope  j/  would  still  be  found 
froni  (6),  and  it  therefore  a])pears  that  the  third  coordinate  of  the  lineal 
elements  of  such  a  family  of  curves  is  expressible  in  terms  of  .'•  and  //. 
In  the  moi'c  general  case  whei'e  the  family  of  curves  is  given  in  tlie 
unsolved  form  F{.i\  //,  < ')  =  0,  tlie  slope  7>  is  found  l)y  the  same  formula 
but  it  now  depends  api)arently  on  C  in  addition  to  on  ,/•  and  >j.  If,  how- 
ever, the  constant  <'  be  eliminated  from  the  two  eijuations 

l'\-';  Ih  ^ ')  =  0     and     ^  +  ^  /.  =  0,  (7) 

there  Avill  arise  an  c(juation  <&(.'',  //,  y/)  =  0  which  connects  tlie  slope  // 
of  anv  curve  of  tlie  family  with  the  coordinates  (.'•,  //)  of  any  point 
tlirough  which  a  cui've  of  the  family  ]iasses  and  at  which  the  slope  of 
that  (airve  is  y.  Hence  it  ajipears  that  the  three  cor.i-'linates  (.'■.//.//)  of 
the  lineal  elements  of  all  the  curves  of  a  family  are  connected  liy  an  eijua- 
tion  4>(.'-.  //,  /')=  0.  just  as  the  corirdinatcs  {.••.  ij.  :.)  of  the  jioints  of  a 
surface  are  connected  by  an  (Miuation  fl^.'',  //,  ,'.)  =  0.  As  the  e(piation 
<!>(./■,  //,  ,'.■)  =  ()  is  called  the  e(juation  of  the  surface,  so  the  e([uation 
<!>(,'•,  //,  ji)  =  0  is  called  the  eipiation  of  the  family  of  cui'ves  :  it  is.  how- 
ever, not  the  finite  e(piatioii  /•''(.;•,  //,  (')  =  0  but  the  diffi'rential  equation 
of  the  family,  because  it  invol\-es  the  di-rivativi'  j/  =  '////'/.','  of  //  by  ,r 
instead  of  the  ]>aranieter  '". 


GENERAL   IXTKODUCTIOX  193 

As  an  example  of  tlie  elimination  of  a  constant,  consider  the  case  of  tlie  parabolas 

I/'-  =  Cx     <iv     if-/x  =  C. 

The  differentiation  of  tlie  equation  in  the  second  form  pves  at  once 

-  ,'/-A-  +  2  yp/x  -  0     or     y  =  2  j-p 

as  the  differential  ecjuation  of  the  family.    In  the  unsolved  form  the  work  is 

2  lip  =  C,         ^-  =  2  ypj,         y  =  2xp. 

The  result  is.  of  cour.se,  the  .same  in  either  case.  For  the  family  here  treated  it 
makes  little  difference  which  method  is  followed.  As  a  general  rule  it  is  perhaps 
best  t(i  solve  for  the  constant  if  the  .solution  is  simple  and  leails  to  a  simple  form 
of  the  f unction /( J",  y)  ;  whereas  if  the  solution  is  not  simple  or  the  form  of  the 
function  is  complicated,  it  is  best  to  ilift'ereutiate  lirst  because  the  differentiated 
equation  may  be  simpler  to  s<ilve  for  the  constant  than  the  original  ecjuation,  or 
because  the  elimination  of  the  constant  between  the  two  eiiuations  can  be  con- 
ducted advantanetnisly. 

If  an  equation  4>  (.'■,  ij. p)  =  0  eonnectinc,'  the  three  coordinates  of  the 
lineal  element  he  i;iven,  the  elements  Avhich  satisfy  the  e(|Uation  may 
l)e  ])lotte(l  much  as  a  sui-face  is  plottt'd ;  tliat  is,  a  pair  of  values  (.'•,  //) 
may  be  assumed  and  sul)stituted  in  the  e(iuation,  tlie  equation  may  then 
he  solved  for  oiu*  or  moi'e  values  of  />,  and  lineal  elements  Avith  these 
values  of  J)  nuiy  l»e  drawn  through  the  jioint  (./■,  //).  In  this  maimer  the 
elements  throiinh  as  many  points  as  dcsii'ed  may  lu-  found.  The  de- 
tached elements  arc  of  interest  and  siynihcance  chit-fly  from  the  fact 
that  thev  can  l)e  (isscmlili'd  into  rm-ri's, — in  fact,  into  the  curves  of  a 
family  F(x,  y,  r)  =  0  of  Avhich  the  ecpiation  $(./■,//,  //)  =  0  is  the  differ- 
ential equation.  This  is  the  converse  of  the  problem  treated  above  and 
re(|uires  the  iiite.n'ration  of  the  differential  e(|uation  <!>(.'•,  //,  /')  =  0  for  its 
solution.  In  some  simple  cases  the  assembling  may  l)e  accomplished 
intuitively  from  the  geometric  iiropcrties  implied  in  the  equation,  in 
other  cases  it  follows  from  the  integration  of  the  equation  by  analytic; 
means,  in  other  cases  it  can  be  done  only  approximately  and  by  methods 
of  computation. 

As  an  example  of  intuitively  assemblini;-  the  lineal  elements  into  curves,  take 

-,   ->         -,        ->       r.                          V /•■-  —  )/- 
*  (•'■■  './■  P)  =  l/'P'  +  '/'  —  '"'  =  0     fJi"    P  =  =: ^  ■ 


The  quantity  V/'-  —  //-  may  be  interpreted  as  one  les  of  a  riirht  triansle  of  which 
y  is  the  other  lei:  and  r  the  hypotenuse.  The  .slope  of  the  hypotenu.se  is  then 
±  y/\'  r-  —  //-  according  to  the  position  of  the  figure,  and  the  differential  equation 
'i>  {.r.  y.  p)  —  0  states  that  the  coordinate  p  of  the  lineal  element  which  satisfies  it 
is  the  negative  reciprocal  of  this  slope.  Hence  the  lineal  element  is  perpendicular 
to  the  hypoteinisf.  Ir  therefore  appears  that  the  lineal  elements  are  tangent  to  cir- 
<■!,■>  of  r.'jlii;^  r  ilc-ci-ilicd  about  points  of  tlie  ..'--axis.    The  eiiuation  of  tln'se  cirrles  is 


194  DIFFERENTIAL   EQUATIONS 

(,r  —  (')-  +  ?/-  =  (•-.  and  tliis  is  therefore  the  inteijral  of  tlie  differential  e(iuation. 
The  correctness  of  this  inteural  niav  be  cliecked  bv  direct  inteicration.    For 


/)  =  —  =  ± or     — —  =  (U     or     ^  /-  —  //-  =  j  —  C. 

dx  y  Vr-i  -  y- 

86.  In  geometric  problems  wliieh  relate  the  slope  of  tlie  tangent  of  a 
curve  to  other  lines  in  the  figure,  it  is  clear  that  not  the  tangent  but 
the  lineal  element  is  the  vital  thing.  Among  such  problems  that  of  the 
iirtJidridUdl  triijcctdfii's  (or  trajectories  inider  any  angle)  of  a  given  family 
of  curves  is  of  es])ecial  importance.  If  two  families  of  curves  are  so 
related  that  the  angle  at  which  any  curve  of  one  of  tlie  families  cuts 
any  curve  of  the  other  family  is  a  right  angle,  then  the  curves  of  either 
family  ai'e  said  to  be  the  orthogonal  trajectories  of  the  curves  of  the 
other  family.  Hence  at  any  point  (./■,  i/)  at  which  two  curves  belonging 
to  the  different  families  intersect,  there  are  two  lineal  elements,  one 
])elonging  to  each  curve,  which  are  perpendicular.  As  the  slopes  of  two 
perpendicular  lines  are  the  negative  reciprocals  of  each  other,  it  follows 
that  if  the  coiu-dinates  of  one  lineal  element  are  (.'■,  y,  p)  the  coordinates 
of  the  other  are  (.>•,  //,  —  1///) ;  and  if  the  coordinates  of  the  lineal  ele- 
ment (.V,  jj,  p)  satisfy  the  equation  <J>(.'', ;/.  ji)  =  0,  the  coordinates  of  the 
orthogonal  lineal  element  must  satisfy  4>  ('./•,  //,  —  l/j))=  0.  Therefore 
the  rub'  forfinillnfi  thu  orfJioriininI  frdjrctnrlrs  iiftJtc  ri/z-rcs  F{:f,  y,  '")  =  0 
is  fo  p'u'l  p'rsf  till'  (lijfi'rrnfidl  i'i{U(itti>n  $(.'■,  //.  //)  =  0  <>f  tJw  ptmiJi/,  tln'ii 
to  ri'jtidci'  p  1)1/  —  \ ^)  to  finil  tlie  dlffprentuil  i'<iii(it!nn  of  tlw  ortJtogonul 
f"U)ll;/,  ami  finiiU  1/  to  lnt('(jr<iti'  tli't^  cqiijit'ion  topwl  tlie  fn  mlli/.  It  may 
be  noted  that  if  7-'(,v)  =  A  (./•,  //)  -f  iY(:i\  if)  is  a  function  of  ,-.'  =  .r  -|-  /// 
(§  73),  the  families  X{.r,  //)  =  C  and  V{.>\  //)  =  K  are  orthogonal. 

As  a  problem  in  ortlioironal  trajectories  find  the  trajectories  of  the  seniicubical 
parabolas  U  —  Cy''  =  y-.    The  differential  eijuation  of  tiiis  family  is  found  as 

3  {X  -  C)-  =  -2  yp.         X  -  C  =  (I  yp)^-.         (|  ypy-  =  y-     or     |  j>  =  yi. 

This  is  the  differential  equation  of  the  given  family'.  Keplace  p  by  —  l/p  and 
integrate  : 

2  1  3      1  3    1  0    1 

—  -  =  i/s     or     1  +   -))ip  =  0     or     dx  +  -  //">  d//  =  0.     and     x  +     v'  =  C. 
:ip  2  2  8 

Thus  the  differential  eijuation  and  finite  efjuation  of  the  orthoconal  family  are  found. 
The  ciu'ves  look  sometliini:  like  parabolas  with  axis  horizontal  and  vertex  toward 
the  right. 

Given    a   differential   e([ttation    $(./■,   //,  p)  =  0   or,    in    solved    form, 

ji  :=  (f)  (./'.  // )  :  Vllo  linriil  rji'iin'iif  iiffo/-/Is  II  llli'iiDS  t'nr  olitn'ni'inij  ip-ii  jiji  iriiJI  ij 
(tJl'linriiirrlriilhinn    a ji p I'o.il ill n t'lo a    to   fjn'snhifi-iil    ir/u'r/i  juissrs    tliroiKjJl 


GENERAL   IN TRODUCTION 


195 


an  tis.^'ttinitl  !„,'int  ^^(•'■,j,  y^-  Eor  the  value  y/^  oi p  at  this  point  may  be 
ct)iiijiuri-il  from  the  equation  and  a  lineal  element  1\J^^  ii^'*-}'  ^^e  drawn, 
the  length  being  taken  small.  As  the  lineal  element  is  tangent  to  the 
curve,  its  end  jjoint  will  not  lie  upon  the  curve  but  will  de}>art  from  it 
by  an  infinitesimal  of  higher  order.  Next  the  slope  ^j^  of  the  lineal 
element  Avhich  satisfies  the  equation  and  i)asses 
throut-h  P,  ]uav  be  found  and  the  element  T\P^ 
may  be  drawn.  This  element  will  not  be  tangent 
to  the  desired  solution  but  to  a  solution  lying  near 
that  one.  Next  the  element  I'J'.^  uiay  be  drawn, 
and  so  on.    The  broken  line  J\J\J'J'.^  ■  •  •  is  clearly 

an  approximation  to  the  solution  and  will  be  a  better  ap])roximation 
the  shorter  the  elements  /',/'( -i  'ii't'  taken.  If  the  I'adiiis  of  curvature 
of  the  solution  at  J\^  is  not  great,  the  curve  will  be  bending  rai)idly  and 
the  elements  must  be  taken  fairly  short  in  (jrder  to  get  a  fair  approx- 
imation ;  but  if  the  radius  of  curvature  is  great,  the  elements  need  not 
be  taken  so  small.  (This  method  of  ap})roximate  graphical  solution 
indicates  a  method  which  is  of  value  in  proving  by  the  method  of 
limits  that  the  equation//  =  (^(.'•,  //)  ac-tually  has  a  solution  ;  but  that 
matter  will  not  be  treated  here.) 

Let  it  be  required  to  plot  approximately  that  solution  of  ?/p  +  x  —  0  which 
pa.'^ses  throuirh  (0,  1)  and  thus  to  tind  the  ordinate  for  j  =  0.5.  and  the  area  under 
the  curve  and  the  lenuth  uf  tlie  curvf  to  tliis  point.    Instead  of  assuminii;  the  lengths 
of  the  successive  lineal  elements.  1ft  the 
lenirths  (jf  successive  increments  5.f  <.if 
X  he  taken  as  5.f  =  0.1.     At   tlic  start 
j-|j  =  0.  (/,j  =  ],  and   fruni  p  =  —  ./■///  it 
f(jllo\vs  tliat  7>,j  =  0.    The  increment  oi/ 
of  //  acijuired  in  movini:-  alnnii'  the  tan- 
gent is  5y  =  p5£  =  0.     Hence  the  new 
point  of  departure  (.r,.  //^)  is  (0.1.  1)  and 
the  new  slope  is  p^  =  —  .i\/ij^  =  —  0.1. 
The  results  of  tlie  W(jrk.  as  it  is  contin-  ' 

ued.  may  be  grouped  in  the  tal)h-.  lli-ncf  it  ajjpcars  that  the  tinal  ordinate  is 
y  =  O.ltO.  By  adding  up  the  trapezoids  the  area  is  compute<l  as  0.48.  and  by  tind- 
imr  the  elements  5s  —  v  5./;-  +  oy-  the  length  is  found  as  O.ol.  Now  the  particular 
equation  here  treated  can  be  inteiir;Ueil. 


1     . 

5x 

5// 

.<■«■ 

Hi 

Pi 

0 

0. 

1.00 

0. 

i  1 

0.1 

0. 

0.1 

1.00 

-0.1 

2 

0.1 

-  0.01 

0.2 

0.09 

-0.2 

8 

0.1 

-  0.02 

0.3 

0.07 

-  o.:n 

4 

0.1 

-  O.O:] 

0.4 

0.04 

-  0.43 

•3 

0.1 

-  0.04 

0.5 

0.90 

yp  +  .f  =  0.  ydy  -I-  xdx 


0. 


.;•-  -{■  y-  =  ( ',     and  hence     x-  -\-  y-  =  \ 


is  the  solution  which  passes  throuuh  (0.  1).  Tlie  ordinate,  area,  and  length  found 
from  the  curve  are  therefore  0.87.  0.48.  0.52  respectively.  The  errors  hi  the 
approximate  results  to  two  places  are  therefore  respectively  o.  0.  2  percent.  If  5x 
had  Ijeeii  chosen  as  0.01  and  four  places  had  been  kept  in  the  computations,  the 
errors  wouM  have  been  smaller. 


196  DIFFEREXTIAL  EQUxVTIOiS^S 

EXERCISES 

1.  In  the  following  cases  eliminate  the  constant  C  to  find  the  differential  equa- 
tion of  the  family  given  : 


(a)  x'  =  2  6V  +  C•^  (/3)  II  =  Cx  +  Vl  -  f^ 

(7)  X-  —  y^  =  C'x,  (5)   ij  =  X  tan  {x  +  C), 

a-  —  C      h'^  —  ('  \dx/  xij  dx 

2.  Plot  the  lineal  elements  and  intuitively  assemble  them  into  the  solution  : 

(a)   lip  +  X  =  0,         (/3)  xp  -  y  =  0,         (7)  r  '-^  =  1 . 

Check  the  results  by  direct  integration  of  the  differential  e(iuations. 

3.  Lines  drawn  from  the  points  {±  (\  0)  to  the  lineal  element  are  equally  in- 
clined to  it.  Show  that  the  differential  equation  is  that  of  Ex.  1  (e).  What  are  the 
curves '? 

4.  The  trapezoidal  area  under  the  lineal  element  equals  the  sectoi'ial  area  formed 
by  joining  the  origin  to  the  extremities  of  the  element  (disregarding  inlinitesinials 
of  higher  order),  (a)  Find  tlu^  (inferential  equation  and  integrate.  {(3)  Solve  the 
same  problem  where  the  areas  are  equal  in  magnitude  but  opposite  in  sign.  What 
are  the  curves  :' 

5.  Find  the  orthogonal  trajectories  of  the  following  families.   Sketch  the  curves. 

(a)  parabolas  y-  =  2  Cx,  Ans.  ellipses  2x-  -|-  y-  =  C. 

(^)  exponentials  //  =  (V'-"',  An><.  parabolas  J  ky-  +  x  =  (.'. 

(7)  circles  (j*  —  C)-  +  //-  =  «-,  Ans.  tractrices. 

(5)  .f^  -  r  =  ('-,       (e)  *"//-  =  .r%       (f)  x'  +  //!  =  ri. 

6.  Show  from  the  answer  to  Ex.  1  (e)  that  the  family  is  self-orthogonal  and 
illustrate  with  a  sketch.  From  the  fact  that  the  lineal  element  of  a  parabola  makes 
equal  angles  with  the  axis  and  with  the  line  drawn  to  the  focus,  derive  the  differ- 
ential ecjuation  of  all  coaxial  confocal  parabolas  and  show  that  the  family  is  self- 
orthogonal. 

7.  If  <!' (x,  v/,  p)  =  0  is  the  diiferential  equation  of  a  family,  show 

/  p-  m  \  1     ^  /  P  +  >"\       ^ 

*   X,  II, =  0     and     4>   .r,  i/,  1  =  0 

\      '     1  -I-  nip/  \      •     1  _  „ii>/ 

are  the  differential  equations  of  the  family  wliose  curves  cut  those  of  the  given 
family  at  tan-i  //(.    Wiiat  is  the  dilference  between  these  two  cases '.' 

8.  Show  that  the  diiferential  ei (nations 


^('J^'L,r,^\  =  0     and     *  (  -  r-^",'^,  r,  0^  =  0 


\'l<p      '7  \  di 


(h^liue  orthogonal  families  in  ])olar  eoiirdiiiatt's,  and  write  the  ecjuation  of  the  family 
which  cuts  the  tii'st  of  these  ai  the  constant  angle  tan-'//). 

9.    Find  the  orthogonal  trajeeturies  of  the;  following  families.    Sketch. 

(((-)    ;■  ^  i(''l>,  (li)  r  =  ('(]  —  cos<;fe),  (7)   /•  =  C'</),  (5)   r-  =  (.'-  cos  2  0. 


GEXEEAL   IXTRODUCTIOX  197 

10.  Recompute  the  approximate  solution  of  yp  +  x  =  0  under  the  conditions  of 
the  text  but  witli  5x  =  0.05,  and  carry  the  work  to  three  decimals. 

11.  Plot  the  approximate  solution  of  p  =  xij  between  (1,  1)  and  the  ?/-axis.  Take 
5x  =  —  0.2.  Find  the  ordinate,  area,  and  lenyth.  Check  by  integration  and 
comparison. 

12.  Plot  the  approximate  solution  oi  j)  =  —  x  through  (1,  1).  taking  5x  =  0.1  and 
following  the  curve  to  its  intersection  with  tlie  .r-axis.  Pind  also  the  area  and  the 
length. 

13.  Plot  the  solution  of  p  =  Vx-  +  </-  from  the  point  (0,  1)  to  its  intersection 
with  the  X-axis.    Take  5x  =  —  0.2  and  find  the  area  and  length. 

14.  Plot  the  solution  of  p  =  .s  which  starts  from  the  origin  into  the  first  quad- 
rant (.s  is  the  length  of  the  arc).  Take  Sx  =  0.1  and  carry  the  work  for  live  steps 
to  find  the  linal  ordinate,  the  area,  and  the  length.    Compare  with  the  true  integral. 

87.  The  higher  derivatives  ;  analytic  approximations.  Although  a 
ditt'erential  equation  4>(.r,  >/,  //')=()  does  not  determine  the  relation 
between  ./■  and  //  Avithont  the  a})[)li('ation  of  some  process  equivalent  to 
integration,  it  docs  afford  a  means  of  computing  the  higher  derivatives 
simply  1)}'  differentiation.    Thus 

((J-  C.r  Cij  '  Of  ' 

is  an  equation  which  may  he  solved  for  y"  as  a  function  of  r,  ?/,  y' ; 
and  //"  may  thei'efore  be  expressed  in  terms  of  .r  and  //  by  means  of 
4>(./-,  //,  if)  =  0.    A  further  ditferentiation  gives  the  e(puition 

ax-       C.I-         c-''Cij  '^•''^il  ^'f  ^'J'^tl 

Cr(b  C^  ?<J> 

Oif-  ^  ClJ-^  CI/'  -^  ' 

which  may  be  solved  for  //'"  in  terms  of  ./',  //.  //',  _?/";  and  hence,  by  the 
preceding  results,  y'"  is  ex})ressil)li'  as  a  function  of  .>■  and  ij  \  and  so 
on  to  all  the  higher  derivatives.  In  this  way  any  property  of  the  inte- 
grals of  <&(.'',  //.  //')  =  0  Avliich,  like  the  radius  of  curvature,  is  exi)ressi- 
ble  in  terms  of  the  derivati\es,  may  be  found  as  a  function  of  ./•  and  //. 
As  the  differential  e(ptation  <5(.'',  //,  //')  =  0  defines  //'  and  all  tlie 
higher  derivatives  as  functions  of  ./■,  //.  it  is  clear  that  the  values  of  the 
derivatives  may  be  found  as  //',,  //".  //,',",  •••  ;it  any  given  point  (,/•.,  ij.y 
Hence  it  is  })Ossible  to  wiite  the  series 

y  =  !/,  +  .<  (:''  -  •'•o)  +  ■>  .'/::  C^  -  -''o)'  +  };  >l7  {■'•  -  a-o)'  +  •  ■  ■•  (^) 
If  this  ]K)wcr  scries  in  ./•  —  ./'.^  converges,  it  defines  y  as  a  function  of 
X  for  values  of  ,/•  near  .'v  :   it  is  inch'cd  tJic   Tuijlur  deci'hipitn'nt  nf  tlie 


198  DIFFEEEXTIAL  EQUATIONS 

function  u  {%  107).    The  convergence  is  assumed.    Then 

y'  =  2/0  +  th  (-^  -  -''o)  +  \  Ih  (•'■  -  ■'"o)'  H — • 
It  may  be  shown  that  the  function   ij  detinetl  by  tlie  series  actually 
satisfies  the  differential  e(|uation  ^{.'•,  //,  ij')  =  0,  that  is,  that 

n  (..■)  =  c^  [,.',  //,  +  'J.  (■'■  -  -''o)  +  h  //u'  ( ■'■  -  -'-of  +  •••,//;  +  //:  (•'•  -  ■''o)  +  •••]  =  0 
for  all  values  of  ./'  near  x^^.  To  prove  this  accurately,  however,  is  beyond 
the  scope  of  the  present  discussion;  the  fact  may  be  taken  for  granted. 
Hence  an  analytic  ex})ansion  for  the  integral  of  a  differential  e(iua- 
tion  has  been  found. 

As  an  example  of  coiuputation  with  liiulier  derivatives  let  it  be  recniired  ti)  deter- 
iiiiiie  the  radius  of  curvature  of  that  solutiou  of  //'  =  tan  (y/Jt)  wliich  pas.ses  tlirouuh 
(1,  1).    Here  the  slope  y^j  j^  at  (1,  1)  is  tan  1  =  1.557.    The  second  derivative  is 

/,     'h'     '^  ,>J     ,  ,0  y  -ry'  -  y 

y    =  —  =  —  tan  -  =  sec- 

(Lr,       dx         X  X      x- 

Froni  these  data  the  radius  of  curvature  is  found  to  be 

11  =  ^l±A!l  ^  sec  ^  -^^  ,         i/o. .)  =  sec  1         ^         =  3.250. 
y"  -t  -iy'  —  y  tan  1  —  1 

The  equation  of  the  circle  of  curvature  may  also  be  found.  For  as  z/^,  j^  is  positive, 
the  curve  is  coiicave  up.  Hence  (1  —  3.2-30  sin  1,1  +  3.2.j0  cos  1)  is  the  center  of 
curvature  ;  and  the  circle  is 

(.c  +  1.735)-  +  (//  -  2.757)-  =  (3.2.50)-. 

As  a  second  example  let  four  tt-i'ms  of  tlie  expansion  of  that  integral  of 
a;  tan  ;/'  =  y  which  pa.sses  throuuh  (2.  1)  be  found.  The  differential  equation  may 
be  .solved  ;  then 

dy      ^       ,  /y\  d'-y      xy'  —  y 

—  —  tan-q  - 


(/,'•  \x!  dx-      ./■'-  +  y- 

iPy  _  (./;-  +  y'-){x  -  1)  y"  +  (3  //-  -  x-)  y'  -  2  xyy'-  +  2  xy 
dx}^  ~  (.f^  +  y-)- 

Xow  it  must  be  noted  that  the  problem  is  not  wholly  determinate  ;  for  y'  is  multi- 
l)le  valued  and  any  one  (if  tlu'  values  f(jr  tan-i  i  may  be  taken  as  the  slope  i.if  a 
solution  through  (2.  1).  Suppose  tiiat  the  aiiule  be  takt-n  in  the  first  quadrant  ;  then 
tan-^  },  =  0.402.  Substitutinu'  this  in  //".  we  find  //,'.',  ,j  =—  0.0152  :  and  hence  may 
be  found  y['.,\^  =  0.110.    The  series  for  //  to  four  terms  is  therefore 

//  =  1  +  0.402  (.;•  _  2)  _  0.0070  (,;•  —  2)-  +  0.018  (./•  —  2)'-. 

It  may  be  notcil  that  it  is  liiMnTally  sinqilcr  not  to  exi)ress  the  hiuher  derivatives  in 
terms  of  X  auil  //.  but  to  cnuqiuie  each  one  successivcl}'  from  the  precediuL:'  nui-s. 

88.  I'icard  has  given  a  nu'tliod  for  tlie  integration  of  the  etjuatiou 
//'  =  </>(•'';  //)  by  s/n-rrssirr  <i jiiii'n.i'i iii'it inns  wliieli.  altliougli  of  the  highest 
llicoretie  value  and  importance,  is  init  jiarlieularly  suitable  to  analytic 


GENERAL   IXTEODUC'TIOX 


199 


uses  in  finding  an  approximate  solution.  The  method  is  this.  Let  the 
equation  _y'  =  c^(./-,  ij)  be  given  in  solved  form,  and  suppose  (./;^,  ij^^)  is 
the  point  through  which  the  solution  is  to  pass.  To  find  the  first 
approximation  let  //  he  held  constant  and  equal  to  _y^,  and  integrate  the 
equation  ij  =  (f>(.'',  //,,).    Thus 

di/  =  cf,  ( ./•,  //J  dx  ;  ij  =  //,,  -{-  4>  (.r,  y^)  d.r  =  J\(.r),  (9) 

where  it  will  l)e  noticed  that  the  constant  of  integration  has  been  chosen 
so  that  the  curve  passes  through  (.z'^,  yj.  For  the  second  approximation 
let  1/  have  the  value  just  found,  substitute  this  in  <j>  (,/•,  _?/),  and  integrate 
auain.    Then 


1/  =  y. 


0  +  j^  <^  ■'-, .%  +£  ^ 


(/':i/^(^^ 


-fp-)- 


(9') 


With  this  new  value  for  y  continue  as  Ix'fore.  The  successive  deter- 
minations of  y  as  a  function  of  .<■  actually  converge  toward  a  limiting- 
function  which  is  a  solution  of  the  e<juation  and  Avhich  passes  throiigh 
(./•|.j,  y^).  It  may  be  noted  that  at  each  ste})  of  the  Avork  an  integration 
is  required.  The  difficulty  of  actually  perfoi'ming  this  integration  in 
formal  practice  limits  the  usefulness  of  the  method  in  such  cases.  It  is 
clear,  however,  that  with  an  integrating  machine  such  as  the  Integra] )h 
the  method  could  be  applied  as  rapidly  as  the  curves  <j){.r, /](./■) )  could 
be  plotted. 

To  see  how  the  method  works,  consider  tlie  intei^ratiou  of  >/  =  :/:  +  //  to  find  the 
integral  through  (1,  ]).    For  tlie  first  approximation  y  =  I.    Tlien 


(7//  r=   (.C  +    1)  (/,/•. 


.'/ 


I  f-  +  .'• 


./•■-  +  x 


/iGO- 


From  this  vahxe  ot  i/  tiie  next  ai)pi'oximatiiin  may  be  i'uniid.  and  then  still  another  : 


dij  =  [x  +  (l  .'•-  +  X  -  V)]  dx,         !/ 
di/  =  [x+f.,{x)]dx,  tj 


=  ■h-''^  +  Ix'-  +  Ix-  +  i^x  +  .^^. 


In  this  case  there  are  no  difficulties  wliich  would  prevent  any  luunber  of  appli- 
cations i:)f  the  method.  In  fact  it  is  evident  that  if  //'  is  a  polynomial  in  x  and  t/.  the 
result  of  any  number  of  applications  of  the  method  will  be  a  polynomial  in  x. 

The  method  of  triulett'rni'nu'il  coefficients  may  often  be  tMn])loyed  to 
advantage  to  d(n'elop  the  solution  of  a  differential  equation  into  a 
series.  The  result  is  of  course  identical  with  that  obtained  by  the 
application  of  successive  differentiation  and  Taylor's  series  as  alxjve  ; 
the  work  is  sometimes  short(U'.  Let  the  equation  Ije  in  the  form 
y/'  =  <^(.'',  //)  and  assuHU'  an  integral  in  the  lV)rm 

!I  =  U,  +   "l  {■'•  -    -'-o)   +  "-2  (.■'■  -  ^■■of  +  "3  (.''■  -  ^'o)'  +  •  ■  •  (10) 


200  DIFFEPvEXTIAL  EQUATIONS 

TlicMi  <f)(.r,  ?/)  may  also  be  expanded  into  a  series,  say, 

<^  (■';  i/)  =  ^\  +  ^\  (•«  -  «'o)  +  ^h  (^  -  «-o)''  +  -I3  (■^■-  -  ^0)'  +  ■  •  •• 

But  by  differentiating  the  assumed  form  for  1/  we  have 

!/'  =  <'r  +  2  a^  (X  -  .g  +  3  a.^  {x  -  x^f  +  4  «^  {x  -  x^f  +  •  •  • . 

Tlius  there  arise  two  different  expressions  as  series  in  x  —  x^  for  the 
function  ?/',  and  therefore  the  corresponding  coefficients  must  be  etjual. 
The  resulting-  set  of  e(]^uations 


i(,  =  A, 


?>a^  =  A^     4.a^ 


A, 


(11) 


may  be  solved  successively  for  the  undetermined  coefficients  a^,  a.„  c.^, 
(I ^,  ■  ■  •  which  enter  into  the  assumed  expansion.  This  method  is  partic- 
ularly useful  when  the  form  of  the  differential  etjuation  is  such  that 
some  of  the  terms  may  be  omitted  from  the  assumed  expansion  (see 
Ex.  14). 

As  an  example  in  the  use  of  undetermined  coefticients  consider  that  sohition  of 
the  equation  y'  =  Vx'-^  +  'iy'^  which  passes  tlu-ou<;h  (1,  1).  The  exjjansion  will  pro- 
ceed according  to  powers  of  x  —  1,  and  for  convenience  tlie  variable  may  be  changed 
to  (  =  X  —  1  so  that 


dt 


=  -^{t  +  1)-  +  ;:5  tf\         ?/  =  1  +  a^i  +  ((o«-  +  a.S'  +  «4<*  + 


are  the  equation  and  the  assnmed  expansion.    One  expression  f(n'  y'  is 

7/'  =  «j  +  2  aj.  +  ?.  a.P  +  4  a^t^  +  •  •  •  • 
To  find  the  other  it  is  necessary  to  expand  into  a  series  in  t  the  expression 

?/  =  V"(i  +  0-  +  ;mi  +  «i^  +  '^-f  +  «3^¥- 

If  this  had  to  be  done  by  Maclaurin's  series,  nothing  would  be  gained  over  the 
niethod  of  §  87  ;  but  in  tiiis  and  many  other  cases  algebraic  methods  and  known 
expansions  may  be  applied  (§  32).  First  scpiare  y  and  retain  only  terms  up  to  the 
third  power.    Hence 

y'  =  2  Vi  +  i  (1  +  -^  "1)  ^  +  -1(1  +  ^  "■!  +  '''  "0  ^'  +  2  ('h<'-2  +  "-.i)  ^•'- 
Now  let  the  (juantity  under  the  I'adical  be  called  1  +  /t  and  expand  so  that 

y'  =  2  Vl  +  h  =  2  (1  +  J  /i  -  I  Ifi  +  tV  '''■')• 
Finally  raise  h  to  the  indicated  powers  and  collect  in  pt)wers  of  t.    Then 


2/'  =  2  +  \  (1  +  3  a,) 


t- 


-  /,.  (1  +  •i",)n  +(><(., +  "^"{) 


GENERAL   INTEODUCXIOX  201 

Hence  the  successive  equations  for  determining  tlie  coefficients  are  a^  =  2  and 

2  «^  =  ^  (1  +  3  «j)  or  a.,  =  |, 

3  rt3  =  i  (1  +  (5  a.,  +  3  a{)  -  Jj  (1  +  3  a^)"  or  a^  =  |-5, 

4  «,  =  I  (rtjo,  +  a.,)  -  J.  (1  +  3  a^)(l  +  0  a,  +  3  a^)  +  ^\  (1  +  3  a,f  or  a^  =  !«-. 

Tlierefore  to  five  terms  tlie  expansion  desired  is 

y  =  1   +  2  (X  -  1)  +  4  (^  _  1)2  +  l|(^  _  1)3  +   ,  M  (X  _  1)4. 

The  methods  of  developing  a  solution  hy  Taylor's  series  or  by  un- 
determined coefficients  apply  ecjually  well  to  equations  of  higher  order. 
Eor  example  consider  an  equation  of  the  second  order  in  solved  form 
y"  =  (f,  (.r,  ]/,  ?/')  and  its  derivatives 

^  ex         CiJ  ^  CiJ  ^ 

''  CX'  CXC(J  ^  cxcij  -^  Cif  ''  CijCtJ  "^  -^ 

C//-  ^  Cij  C;/ 

Evidently  the  higher  derivatives  of  y  may  be  obtained  in  terms  of  x, 
y,  y' ;  and  y  itself  may  be  written  in  the  ex}»anded  form 

where  any  desii'ed  values  may  be  attributed  to  the  ordinate  //^  at  which 
the  curve  cuts  the  line  x  ■=  ./■,,  and  to  the  slope  //'j  of  the  curve  at  that 
point.  ]\roreover  the  coeffic-ients  //,",  y'^\  ■  ■  ■  are  determined  in  such  a  way 
that  they  depend  on  the  assumed  values  of  //,,  and  y',,.  It  therefore  is 
seen  that  the  solution  (12)  of  the  difterential  (-(piation  of  the  second 
order  rt^ally  involves  two  arbitrary  constants,  and  the  justitication  of 
writing  it  as  F(x,  y,  C^,  ('J  =  0  is  clear. 

In  following  out  the  method  of  undetermined  coefficients  a  solution 
of  the  equation  would  be  assmued  in  the  form 

U  =  !/o+  !/'>(:■■  -  ^0  +  <',(:•■  '  ■'■.)"  +  ",(:■'  "  -^'o)'  +  ^'S''  "  ^'o)'  +  ' '  " .  (13) 
from  which  y'  and  y"  would  l)c  oljtained  by  differentiation.  Then  if  the 
series  for  y  and  //'  be  sul)stituted  in  //"  =  c/)(.'',  y,  y')  and  the  result 
arranged  as  a  series,  a  second  expression  for  //"  is  obtained  and  the 
comparison  of  the  coefficients  in  the  two  series  will  afford  a  set  of  equa- 
tions from  which  the  successive  coeffi(-ients  may  be  found  in  terms  of 
//,,  and  /^  b}'  solution.  These  results  may  clearly  be  generalized  to  the 
case  of  differential  e(piations  of  the  ?(th  order,  whereof  the  solutions 
will  dep(Mid  on  n  arbitrary  constants,  namely,  the  values  assumed  for 
a  and  its  first  n  —  1  derivatives  when  ./■  =  ,/■.. 


202  DIFFERENTIAL  EQUATIONS 

EXERCISES 

1.  Find  tlif  radii  and  circles  of  curvature  of  the  solutions  of  the  followin<r  equa- 
tifjus  at  the  points  indicated  : 

(a)   >/  =  Vx-  +  !/-  at  (0,  1).  (/3)  yy'  +  x  =  0  at  {x^,  y^). 

2.  Find  !/[[[  J)  =  (5  V2  -  2)/4  if  /  =  Vx'  +  y-. 

3.  Given  the  equation  y-y''^  +  xyy"~  —  yy'  +  x-  =  0  of  the  third  degree  in  y'  so 
that  there  will  be  three  solutions  with  different  slopes  through  any  ordinary  point 
(x,  y).    Find  the  radii  of  curvature  of  the  three  solutions  through  (0.  1). 

4.  Find  three  terms  in  the  expansion  of  the  solution  of  y'  —  (-"'  about  (2.  i). 

5.  Find  four  terms  in  the  expansion  of  the  solution  of  (/  =  logsinx(/  abmit  {\  tt.  1).. 

6.  Ivxpand  the  solution  of  y'  =  xy  alxmt  (1.  y^^  to  five  terms. 

7.  Expand  the  solution  of  ;/'  =  tan  (///x)  about  (1.  0)  to  four  terms.  Note  that 
here  x  should  be  expanded  in  terms  of  y.  not  y  in  terms  of  x. 

8.  Expand  two  of  the  solutions  of  y-y'"^  +  xyy"-  —  yy'  -|-  x-  =  0  about  (—  2,  1) 
to  four  terms. 

9.  ( )btain  four  successive  aiiproximations  to  the  integral  of  y'  —  xy  through  (1, 1). 

10.  Find  four  successive  aiiproximations  to  the  integral  of  y'  =  x  ■\-  y  through 

(0.  y,). 

11.  Show  by  successive  approximations  that  the  integral  of  y'  =  y  through  (0,  y^,) 
is  the  well-known  y  =  y„e^'. 

12.  Carry  the  approximations  to  the  solution  of  //'  =  —  x/y  through  (0,  1)  as 
far  as  you  can  integrate,  and  plot  each  approximation  on  the  same  figure  with  the 
exact  integral. 

13.  Find  by  tlie  method  of  undetermined  coefficients  the  number  of  terms  indi- 
cated in  the  expansions  of  the  solutions  of  these  differential  equati(jns  about  the 
points  given  : 

(a)  y'  —  Vx  +  //,  five  terms,  (0,  1),        (^)  y'  —  x'x  +  //,  four  terms,  (1,  3), 
(7)   il'  =  ■>'  +  ,'/.  "  terms,  (0.  //,,).  (5)  //'  =  x 'x-  +  y-.  four  terms,  (f.  \). 

14.  If  the  solution  of  an  equation  is  to  be  expanded  about  (0.  //,,)  and  if  tlie 
change  of  x  into  —  x  and  y'  into  —  //'  does  imt  alter  the  (Mjuation.  the  s(.jlutiiin  is 
necessarily  synunctric  with  respect  to  the  y-axis  and  the  expansion  may  be  assumed 
to  contain  only  even  powers  of  x.  If  tlie  solution  is  to  be  expanded  about  (0.  0) 
and  a  change  of  ,/•  into  —  x  and  y  into  —  y  does  not  alter  the  I'quation.  the  solution 
is  synunctric  with  respect  to  the  ori;:in  and  the  expansion  may  be  assumed  in  odd 
powers.  Obtain  the  expansions  to  four  terms  in  the  following  cases  and  compare 
the  lal)or  invojvcil  in  the  method  of  undetermined  coefficients  with  that  whic'h 
would  lie  involved  in  jiei-foi'ming  the  retjuisite  six  or  seven  differentiations  f(jr  the 
ajiplication  of  .Maclauriifs  series: 

(n-)   y'  = about  (0.  2),  (/i)  y'  -  sin  xy  about  (0.  1), 

A 'x- +//■-' 
(7)  y'  =  e'J  about  (0.  0).  (5)  //'  =  x^y  +  xy-'  about  (0.  0). 

15.  Expand  to  and  including  the  term  x''  : 

(a)   y"  —  //'-  +  ,(■//  about  x,,  =  0.  y^^  =  n^^,  //,'  —  n ^  (liy  both  methods). 
(.i)  xy"  -f  ;/'  -L  7  —  0  iiliiiut  x^  =  (t.  y^^  =  a^^.  y[^  =  —  "^  (by  und.  coeffs.). 


CHAPTER  VIII 

THE  COMMONER  ORDINARY  DIFFERENTIAL  EQUATIONS 

89.  Integration  by  separating  the  variables.  If  a  differential  equa- 
tion of  the  first  order  may  Ije  solved  for  //'  so  tliat 

l,'  =  ^(x,y)      or      M{.r,  >, )>!,■  +  X{x,y)dy  =  0  (1) 

(where  the  functions  <^,  M,  X  are  single  valued  or  where  only  one  spe- 
cific branch  of  each  function  is  selected  in  case  the  solution  h^ads  to 
multiple  valued  functions),  the  differential  equation  involves  only  the 
first  power  of  the  derivative  and  is  said  to  l)e  of  the  first  degree.  If, 
furthermore,  it  so  happens  that  the  functions  <^.  -1/,  .V  are  products  of 
functions  of  x  and  functions  of  //  so  that  the  equation  (1)  takes  the  form 

I/'  =  4>^(.r)  <f>Ji/)      or     M^(.r)  M.i>,)  d:r  +  X^-)  Xjj/ )  ./y  =  0,         (2) 

it  is  clear  that  the  variables  may  be  separated  in  the  manner 

and  the  integration  is  then  immediately  performed  l>y  integrating  each 
side  of  the  equation.  It  was  in  this  way  that  the  numerous  problems 
considered  in  Chap.  VII  were  solved. 

As  an  example  ccmsider  the  equation  yy'  -\-  xy-  =  x.    Here 

ydii  +  X  {y-  -  1)  dx  -  0     or     J^-JL.  +  ^rh  =  0, 

and  h  log  {y-  —  I)  +  \  x-  =  C     or     {y-  —  1)  c'-  =  C. 

The  second  form  of  tlie  solution  is  found  by  taking  the  exponential  of  both  sides 
of  the  first  form  after  multiplying  by  2. 

In  some  differential  equations  (1)  in  which  the  varialjles  are  not 
immediately  separable  as  alcove,  the  introduction  of  some  change  of 
variable,  whether  of  the  dependent  or  independent  varial)le  or  both, 
may  lead  to  a  differential  equation  in  Avhich  the  new  varial)lt's  are  se])a- 
rated  and  the  integration  may  be  accomplished.  The  selection  of  the 
proper  change  of  variable  is  in  general  a  matter  for  the  exercise  of 
ingenuity  :  succeeding  paragraphs,  however,  will  point  out  some  special 

203 


204  DIFFERENTIAL    EQUATIONS 

types  of  equations  for  which  a  definite  type  of  substitution  is  known 
to  accomplish  the  se})aration. 


As  an  example  consider  the  e(iviati()ii  xAy  —  ydx  =  x  Vx-  +  y-  dx,  where  the  varia- 
bles  are  clearly  not  separable  without  substitution.  The  presence  of  Vx-  +  y'"^ 
suggests  a  change  to  polar  coordinates.    The  work  of  finding  the  solution  is  : 

X  =  r  cos  0,     y  =  r  sin  ^,     dx  =  cos  ddr  —  r  sin  0d9,     dy  =  sin  Bdr  +  r  cos  Odd  ; 

then  xdy  —  ydx  =  r-dO,         x  Vx-  +  y'-dx  =  r-  cos  dd  (r  cos  0). 

Hence  the  differentia)  e(iuation  may  be  written  in  the  form 

f-dd  =  r-  cos  Od  {r  cos  $)     or     sec  ddO  =  d  {r  cos  8), 

and  los;  tan  (J  0  +  W)  =  r  cos  0  +  C     or     log  — ~ '—  =  x  +  C. 

cos  6 


y  x'^  _i_  ^/-  _i_  y 

Hence  — '- ^ ^  =  Ce''         (on  substitution  for  ff). 

X 

Another  change  of  variable  which  works,  is  to  let  y  =  vx.    Then  the  work  is ; 


x{vdx  +  xdv)  —  vxdx  =  x-  a/  1  +  v'-dx  or  du  =  V 1  +  t-dx. 

dv 
Then  ,  =  dx,         sinh-iu  =  x  +  C,         y  =  x  sinh  (x  +  C). 

Vl  +  V- 

This  solution  turns  out  to  be  sliorter  and  the  answer  appears  in  neater  form  than 
before  obtained.  The  great  difference  of  form  that  maj'  arise  in  the  answer  when 
different  methods  of  integration  are  employed,  is  a  noteworthy  fact,  and  renders  a 
set  of  answers  practically  worthless  ;  two  solvers  may  frequently  waste  more  time 
in  trying  to  get  their  answers  reduced  to  a  common  form  than  each  would  spend  in 
solving  the  problem  in  two  ways. 

90.  If  in  the  equation  //' =  c/)(.v,  //)  the  fiuiction  <^  turns  out  to  be 
(^  (///./•),  a  function  of  ///,/•  ah)iu'.  that  is,  if  the  functions  M  and  A'  are 
homogeneous  functions  of  .r,  //  and  of  the  same  order  (§  53),  tlie  differ- 
ential equation  is  said  to  be  /ioii)');/t'7ieoi/s  and  the  change  of  varial)le 
1/  =  vx  or  :r  =  r/j  Avill  always  result  in  separating  the  variables.  The 
statement  may  be  tabulated  as  : 

if  ^  =  ci(-),         substitute  I       '/='■•'■  /3) 

(/./•        '^  y./y  [  or  ./."  =  ry. 

A  sort  of  corollary  case  is  given  in  Ex.  6  below. 

As  an  example  take  y\l  +  t'7'ix  +  c'' {y  —  x)dy  —  0,  of  whieli  tlie  liomogeneity 
is  perhaps  somewhat  distjuised.    Here  it  is  better  to  choose  x  =  vy.    Then 

(1  +  C)  dx  +  e'-  (1  —  t)  dy  =  0     and     dx  =  vdy  +  ydv. 

Hence  (u  +  t') dy  +  y (1  +  e) dv  =  0     or     —  +  -        '    du  =  0. 

y       r  +  c'- 

Hence  log  y  +  log  (f  +  c')  =  C    or     x  +  yc''  =  C. 


COMMONER  OEDIXARY   EQUATIONS  20-3 

If  the  differential  equation  may  be  arranged  so  tliat 

%  +  -^'i(-^-)  U  =  '^'X-'-)  'f     o^-     ^  +  y.iu)  ^r  =  y/jj)  .r",  (4) 

where  the  second  form  diifei'S  from  the  first  only  through  the  inter- 
change of  .r  and  >/  and  wliere  A'^  and  A'.,  are  functions  of  .'•  alone  and 
I'j  and  }\  functions  of  //,  the  equation  is  called  a  BernoaHl  ciiiKithtii :  and 
in  particular  if  ?i  =  0,  so  that  the  dependent  variable  does  not  occur  on 
the  right-hand  side,  the  equation  is  called  Ibiefw.  The  substitution 
which  separates  the  variables  in  the  respective  cases  is 

y  =  rr-/-*'>('>'''-     or     .r  =  /•«-/^'i^-"^''-".  (5) 

To  show  that  the  separation  is  really  accomplished  and  to  find  a  general 
formula  for  the  solution  of  any  Bernoulli  or  linear  equation,  the  sui> 
stitution  may  be  carried  out  formally.    For 

dx       dx"  ''^' 

The  substitution  of  this  value  in  the  equation  gives 

<1 1'         r  c  (II'  r 

—  t- J -^V'-  =  A  ,r"c- "/ -^•■''.'-     or     —  =  X/'-  '"/•*■>''.'- ,/,,.. 
dx  '  f" 

Hence  v^^' 


1  -  n)   Cx/'-"Kf-''^'''d.,;      when      >i  ^  i; 


or  v/i-''=  (1  —  «)f>-i'j 


f-v./. 


I  _Y/'<i-"'/-^V'.'- 


(<>) 


There  is  an  analogous  form  for  the  second  form  of  the  equation. 

The  equation  {x-y^'  +  xy)dy  =  dx  may  be  treated  by  this  method  by  writing  it  as 

dx 

yx  —  y"'x-     su  that     Y.  =  —  >/,  Y.^  =  y^,  n  =  2. 


Then 

let 

X  =  re   ^'           =  i-c-    . 

Then 

dx                dr.    \  V-            \  '1-            \  .'/■-      dv    \  v- 

yx  =  —  c-     +  vyc-     —  yvc     =  —  e" 

dy               dy                                          dy 

and 

dv   ly-        „  ,    ,            dv        „  },>fi , 
—  e-      —  yv-c'J-     or     —  =  y"c-    di/, 
dy                                   V- 

and 

1  _ 

V 

--  (y-^  -  2)  e-  ■    +  C     or     -  =  2  -  y'^  +  Ce    "  ' . 

This 

result  could  hav 

e  been  obtained  by  direct  svdjstitution  in  the  fornuda 

^1-  «  =  (1  _  „)  e<"  -'\f  ^V"  r  J  Yj^-  ")/  >V/v  ^jl  ^ 

but  actually  to  carry  the  method  through  is  far  more  instructive. 

*  If  /<  =  !,  the  variables  are  separated  in  tlie  original  equation. 


206  .    JJIFFEIIE^TIAL   EQUATIU^^S 

EXERCISES 

1.  Solve  Uie  equations  (variables  iuiiiiediately  separable)  : 

(a)  (1  +  x)ij  +  {I-  y)sj/  ^  0,  Ann.  xy  =  Ce''--^ 

(/3)  a  {xdy  +  2  ydx)  =  xijdy,  (7)   Vl  —  x-  dij  +  Vl  —  y'^  dx  =  0, 

(5)  (1  +  y-)  dx  -  (//  +  Vl  +  y){\  +  x)3  cZy  =  0. 

2.  Dy  various  ingenious  changes  of  variable,  solve  : 

(a)  (x  +  y^y'  =  a-,  Ans.  x  +  y  =  a  tan  (y/a  +  C). 

(/3)  (x  —  //-)  dx  +  2  zydy  =  0,  (7)  xdy  —  //dx  =  (x^  +  j/''^)  dx, 

(5)  y'  =  X  -  y,  (e)  v/y'  +  r  +  X  +  1  =  0. 

3.  Solve  these  homogeneous  equations  : 

(a)  (2  Vxy  —  x)  y'  +  ?/  =  0,  yl7i.s.   Vx///  +  log  ?/  =  G. 

V 

(/3)  xe''  -V  y  —  xy'  =  0,  ^ns.  y  +  x  log  log  C/x  =  0. 

(7)   (x^  +  y'^)  dy  =  xydx,  (5)  xy'  —y=  Vx'-^  +  y'^. 

4.  Solve  these  Bernoulli  or  linear  equations  : 

(or)  //  +  y/x  =  y~,  Ans.  x// log  Tx  +  1  =  0. 

(/3)  //'  —  //  CSC  X  =  cosx  —  1,  Ans.  y  =  sin  x  +  C  tan  }  x. 

(t)  x//  +  y  =  //■'  logx,  Anx.  //-i  =  logx  +  ]  +  Cx. 

(5)  (1  +  y'^)  dx  =  (tan- 1  y/  —  x)  d//,  (e)  //dx  +  {(ix'-y"-  —  2  x)  d//  =  0, 

(f )  x//'  -  a//  =  X  +  1,  {ri)  yy'  +  i  //-  =  cos  x. 

5.  Show  that   the  substitution  y  =  vx  always  separates  the  variables   in  the 
homogeneous  ecjuation  y'  =  (p  (y/x)  and  derive  the  general  formula  for  the  integral. 

6.  Let  a  differential  equation  be  reducible  to  the  form 

dy  _     /f'lX  +  '>i//  +  cA  a^h.,  —  <i.J)^  ^  0, 


dx         \«^x  +  /a,//  +  cj  ov    (i^h.,  —  <(J)^  =  0. 

In  case  a^h.,  —  aji^  -^  0,  the  two  lines  r/,x  +  h^y  +  r,  =  0  and  fl._,x  +  '>.,//  4-  r^  =  0 
will  meet  in  a  point.  Show  that  a  transformation  to  this  point  as  origin  makes 
tlie  new  equation  homogeneous  and  hence  soluble.  In  case  «,/».,  —  aj>^  =  0,  the 
two  lines  are  parallel  and  the  substitution  z  —  a.,x  +  b.,y  or  z  —  a^c  +  b^y  will 
separate  the  variables. 

7.  By  the  method  of  Kx.  G  solve  the  ecpiations : 

(a)   (?>//  — 7x  +  7)dx  +  ('//—;>x  +  8)d//  =  0,  Ans.   (//_  x +!)-(//+ x— 1)5  =  C. 

(^)   (2x  +  8//-5)//'  +  (3x  +  2//-y)=0,         (7)   (4x  +  3 //+ l)dx+ (x  +  /y+l)d//  =  0, 

(5)   (2x  +  y)  =  //'(4x  +  2  //  -  1),  (e)  '^  =  (t^"^-^)'' • 

dx       \2  X  —  2  //  +  1/ 

8.  Show  that  if  the  equation  may  bo  written  as  yf(xy)dx  +  xg{xy)dy  =  0, 
where /and  g  are  functions  of  the  product  xy,  the  substitution  w  =  x//  will  sepa- 
I'ate  the  variables. 

9.  By  virtue  of  Ex.  8  integrate  the  ecpiations  : 

(a)   {y  +  2x?/2  —  x-y"')dx  +  2 x-ydy  =  0,  Ans.  x  +  x-y  =  (' {^  —  xy). 

(i-i)   (//  +  xy-)  dx  +  (x  -  x~y)  dy  -  0,  (7)   (1  +  xy) xy-dx  +  {xy  -  1 )  xdy  =  0. 


COMMONER   OKDINAKY   EQUATIONS  20T 

10.  By  any  method  that  is  applicable  solve  the  fnllowinp;.  If  more  than  one 
method  is  applicable,  state  what  methods,  and  any  apparent  reasons  for  choos- 
ing one  : 

(«)  V'  -\-  y  cos  X  =  2/»  sin  2cc,  (/3)   (2  xhj  +  3  if)  dx  =  {x^  +  2  xy"-)  dy, 

(y)  {4:x  +  2y  -  l)y'  +  2x  +  y  +  1  =  0,      (5)  yy'  +  xy^  ^  x, 

{e)  y'  sin'?/  +  sin  x  cos  y  =  sin  .r,  (f)   Va"-^  +  x'^  (1  —  ?/')  =  x  +  ?/, 

(t;)   (x^y'^  +  x-?/-  +  x(/  +  1)  y  +  (x'V''  —  x-y-  —  xy  +  I)  xy\        [6)  y'  =  sin  (x  — ?/), 

_  V 

( t )  xydy  —  tf-dx  =  (x  +  y)-  e   ■'■  (7x,  (k)   (1  —  2/-)  dx  =  nxy  [x  +  l)dy. 

91.  Integrating  factors.  If  the  e(iuation  Md.r  +  Xd//  =  0  hy  a  suita- 
ble rearraiigenuMit  of  tlu>  tenns  can  V)e  })iit  in  the  form  of  ii  siun  of  total 
differentials  of  certain  functions  u,  r,  •  •  • ,  say 

da  +  (//;  H =  0,     then      //  +  ^>  +  . .  •  =  (/  (") 

is  surely  the  solntion  of  the  equation.  In  this  case  the  equation  is  called 
an  exact  different Icl  equation.  It  frequently  hapjiens  that  althousj^'h  the 
equation  cannot  itself  be  so  arranged,  yet  the  e(|uation  obtained  from 
it  by  multiplying  through  Avith  a  certain  factor  fi^.r,  //)  may  be  so 
arranged.  The  factor  /x(,r,  v/)  is  then  called  an  Integrat'mg  fa<'to)'  oi  the 
given  equation.  Thus  in  the  case  of  variables  se2)arable,  an  integrating 
factor  is  1/M\^X^ ;  for 

— ^  \M\M,  d.r  +  A,.V,  d>n  =  '^^'^  dx  +  ^^^  di/  =  0  :  (8) 

and  the  integration  is  immediate.  Again,  the  linear  e(|uation  may  be 
treated  by  an  intt'grating  factor.    Let 

d^  +  \\f/dx  =  X,//x     and      /x  =  rPv''-;  (9) 

then  t'/-^'''/.'-  ,/y  4-  A\r/-^V'.'-  y,/,^  =  ^,/.av/x  y^,/,.  ^^q-^ 


(/[//r/-^.''']  =  J-^VZ-AV/.r,     and     yr/-^V'-=    f  e  f  ■''"■■  X  jh 


01) 


In  the  case  of  variables  separable  tlie  use  of  an  integrating  factor  is 
therefore  implied  in  tlie  ])rocess  of  separating  tlie  variables.  In  the 
case  of  the  linear  equation  the  us<^  of  the  integrating  factor  is  somewhat 
shorter  than  the  use  of  the  substitution  for  separating  the  variables. 
In  general  it  is  not  jiossible  to  hit  \q)on  an  integrating  factor  by  ins})ec- 
tion  and  not  practicable  to  obtain  an  integi'ating  factor  by  analysis,  but 
the  integration  of  an  equation  is  so  simi)le  when  the  factor  is  known, 
and  the  equations  which  arise  in  i)ractice  so  frequently  do  have  sinq)le 
integrating  factors,  that  it  is  worth  while  to  examine  the  equation  to 
see  if  the  factor  cannot  be  determined  l)y  ins})ection  and  trial.  To  aid 
in  the  work,  the  dil'fercntials  of  the  simpler  functions  such  as 


208  DIFFERENTIAL   EQUATIONS 

dxy  —  xiJij  -\-  ydx,  \  d  (./•'-  +  i/'-j  =  xdx  -\-  ydij, 

^y^xdji_-ydx^  ^^^vH^V'^'^-^'h/^  ^12) 

X  X-  y         X-  +  y 

should  he  home  in  mind. 

Consider  tli(3  eciuatioii  (x*6-^  —  2  ?/«//-)  dx  +  2  7?ix-;/d// =  0.  Here  the  first  term 
x^e^dx  will  be  a  diffeiTutial  of  a  function  of  x  no  matter  what  function  of  x  may  be 
assumed  as  a  trial  )x.    With  /u  =  ]//■*  the  eciuation  takes  the  form 

edz  +  2  ?/i    ■-  ■   - =  de-  +  hid  —  =  0. 

\  X-  x^  /  X- 

The  integral  is  therefore  seen  to  be  e''  +  my-/x"  —  ('  without  more  ado.  It  may 
be  noticed  that  tliis  e(iuatiou  is  of  the  Bernoulli  type  and  that  an  integration  by 
that  method  would  be  considerably  longer  and  more  tedious  than  this  use  of  an 
integrating  factor. 

Again,  consider  (x  +  i/)dx  —  (x  —  y)di/  —  0  and  let  it  be  written  as 

xdx  +  7jdy  +  ydx  —  jcdij  =  0  ;     try     fj.  =  l/(x-  +  ?/-) ; 

xfZx  +  ydy      yds,  —  xdi/      ^  1  , ,      ,   ,         ,x        ,  ,  x      ^ 

then  -^/ j^  j^  •!_ j_  ^  q     ,^j.     _  ^^  log  (x-  +  y-)  +  d  tan-i  -  =  0, 

X-  +  %f-  X-  +  2/-  2        '  y 


and  the  integral  is  log  Vx-  +  ij^  +  tan-i  (x/y)  =  C  Here  the  terms  xdx  +  ydy 
strongly  suggested  x-  +  y-  and  the  known  form  of  the  differential  of  tan-i  (x/?/) 
corroborated  the  idea.  This  equation  comes  under  the  homogeneous  type,  but  the 
use  of  the  integrating  factor  considerably  shortens  the  work  of  integration. 

92.  The  attempt  has  heen  to  write  Mdx  +  Xdy  or  /i  (.l/r/.r  +  Ar/y) 
as  the  sum  of  total  differentials  du  +  du  +  •  •  • ,  that  is,  as  the  diffei'eiitial 
dF  of  the  function  ?/  +  /'  +  ■•  •,  so  that  tlie  solution  of  tlie  ecjuation 
Mdx-\-Xdy  =  0  (-ould  he  ohtained  as  F=  (\  A\"hen  tlie,  ex})ressi()ns 
are  complicated,  the  attempt  nia}'  fail  in  })ractice  even  where  it  theoi-eti- 
cally  should  succeed.  It  is  therefore  of  importance  to  estahlish  condi- 
tions under  which  a  differential  expression  like  I'dx  +  <■(?'/// shall  lie  the 
total  differential  dF  of  some  function,  and  to  hnd  a  means  of  obtaining 
F  when  the  conditions  are  satished.    This  will  now  he  done. 

(13) 


Sir 

[)pose 

Pdx  +  Qdy  = 

--dF  = 

=  ^  dx  +  .     d>/  ; 
ex             cy 

tlien 

cF 

ex 

cF 

dp       cQ        c-F 
cy        ex       cxcy 

Hence  if  Pdx  +  (^dy  is  a  total  differential  dF.  it  follows  (as  in  §  52)  that 
the  relation  /'J^  =  ''/,.  must  hold.  Now  convci-sely  if  this  relation  docs 
liold,  it  may  he  shown  that  Pdr  +  (idi/  is  the  total  diilerential  of  a 
function,  and  tliat  this  function  is 


(14) 


COMMONER  ORDmARY  EQUATIONS  209 

or  F=   f  \l (,r,  >j) drj  +    f/^ (.r,  ij^ (b; 

where  the  fixed  value  .7-^  or  y^  will  naturally  be  so  chosen  as  to  simplify 
the  integrations  as  much  as  possible. 

To  show  that  these  expressions  may  he  taken  as  F  it  is  merely  neces- 
sary to  compute  their  derivatives  for  identification  with  P  and  Q.   Now 

^  =  ■^1    P(x,  y)dx  ^Yx]  ^^*^'''«'  ^^'^^  =  ^'^'''  y^' 


dF 


'i  ^hj?^^'  '^'^  ^hJ''^'^'  '^''  ^  hJ'^'-'  ^  '^^•'■°'  '^- 


These  differentiations,  applied  to  the  first  form  of  F,  require  only  the 
fact  that  the  derivative  of  an  integral  is  the  integrand.  The  first  turns 
out  satisfactorily.  The  second  must  be  simplified  by  interchanging  the 
order  of  differentiation  l)y  y  and  integration  by  x  (Leibniz's  Rule, 
§  119)  and  by  use  of  the  fundamental  hypothesis  that  P'y  =  Q'^. 

i.  J  /V,«  +  Q(x^,  y)  =j    1^  <lr  +  Q{x^,  y) 

The  identity  of  /'  and  Q  with  the  derivatives  of  F  is  therefore  estab- 
lished.  The  second  form  of  F  Avould  l)e  treated  similarly. 

Show  that  (x-  +  log  y)(lx  +  x/ydy  =  0  is  an  exact  differential  equation  and  obtain 
the  solution.    Here  it  is  first  necessary  to  apply  the  test  P^  =  Q^ .    Now 

—  (x-  +  log  y)  =  -     and =  -  • 

cy  y  cxy      y 

Hence  the  test  is  satisiied  and  the  integral  is  obtained  by  applying  the  formula : 

/■  ■'  r  0  \ 

I     (x-  +  log  y)  dx  -\-  \  "  dy  =  -x-"  +  x  log  y  =  C 
Jo  '  J    y  3 

J->  II  X  p  1 

-  dy  +    I  (x-  -1-  log  1)  dx  =  X  log  y  +  ~  •''■'  —  C. 
1    y    '        J  3 

It  sliould  be  noticed  that  the  choice  of  x,,  =  0  simplifies  the  integration  in  the  first 
case  because  the  substitution  of  the  lower  limit  0  is  easy  and  because  the  .second 
integral  vanishes.  The  choice  of  ?/o  =  1  introduces  corresponding  simplifications  in 
the  second  case. 


Mx 

+ 

N,l 

y-n 

ip- 

-,0/.V 

\<h 

1 

210  J)IFFERENTTAL   EQUATIONS 

Derive  tlie  partial  differential  equation  lohich  any  inter/rating  factor  of  the  differ- 
ential equation  Mdx  +  Ndy  =  0  must  satiffy.    If  fj.  is  an  intef^ratiiig  factor,  tlien 

liMdx  +  ^iNdy  =  dF    ami     -^~—  =  -^-- . 
cy         dx 

Hence  M~  —  N^  =  fil )  (15) 

dy  ex         \dx        cy  I 

is  tlie  desired  equation.  To  determine  tlie  integrating  factor  by  solving  this  equa- 
tion would  in  general  be  as  ditHcult  as  solving  the  original  ecjuation  ;  in  some 
special  cases,  however,  this  equation  is  useful  in  determining  ^. 

93.  It  is  now  convenient  to  tabulate  a  list  of  different  types  of  dif- 
ferential equations  for  whic.li  an  integrating  factor  of  a,  standard  form 
can  be  given.  With  the  knowhidge  of  the  factor,  the  ecpiations  may 
then  be  integrated  by  (14)  or  by  inspection. 

EciUATiox  Mdx  -f  Nilij  =  0  :  Factor  fj, : 

I.  Homogeneous  J\ldx  +  Nd/j  =  0, 
II.  Bernoulli  <///  -\-  X^ijdx  =  A'j/'dx, 

m.  _v  =  ,/■(,,„),  .V  =  ..,(.,,/),  ^^^_^^ 

VI.  Typo  x^ifi^mudx  +  nxd/j)  =  0,  i'',  "'     .  ";^  '\       ' 

VII.   x^/fihnmJx  -}-  vxd //)-{-  xy//^(  p//(/x  -\-  fix(h/)=  0,         <  ' ,    .         '    .  ' 

'  ■        '  l/:  detci'mmed. 

The  use  of  tlie  integrating  factor  oi'tcn  is  simpler  than  the  substitu- 
tion )/  =  rx  in  the  liomogeneous  ('(piation.  It  is  practically  identical 
with  tlie  sid)stitution  in  the  Bernoulli  ty})c.  In  the  third  ty})e  it  is 
often  shorter  than  the  substitution.  The  remaining  ty])es  have  had  no 
substitution  in(li(;ated  for  them.  The  ])ro()fs  that  tlu;  assigned  forms 
of  the  factor  are  right  ai'c  given  in  tlu^  examples  below  or  ai-e  left  as 
exercises. 

To  .show  tliat  /J.  =  {}[x+  \y)^^  is  an  iiifegrating  factor  for  tlic  humogeiieeus 
case,  it  is  possil)lc  sinq)ly  to  suhstitutc  in  tlic  eiiiiatioii  (1^")),  which  /x  must  satisfy, 
and  show  that  Ihc  equation  actually  liohls  by  virtue  of  tlic  fact  that  M  and  .V  are 


a.iA 

cX 

v. 

If 

dx 

dx 
d.U 

-,m, 

V. 

If 

dx 

^// 

--/(!/), 

COMMONER  OEDINARY  EQUATION'S  211 

homogeneous  of  the  same  degree,  —  this  fact  being  used  to  simi)lify  the  result  by 
Euler's  Formula  (30)  of  §  53.    But  it  is  easier  to  proceed  directly  to  show 

M  c  /       N       \  a  /I      1     \       0  /I      0     \         ,  Ni/ 

—  "••  '  —       '  where     </>  = 


dy  Mx  +  Ny      ex  \Mx  +  NyJ  cy\xl  +  (pj      cx  \y  I  -\-  (pj  Mx 

Owing  to  the  homogeneity,  4>  is  a  function  of  y/x  alone.  ^  Differentiate, 
a      1     \  I       (p'       1       I       4)'         —  y       c/1      (fi 


cy  \xl  +  (pf  X  {I  +  0)-  X      y  (1  +  (p)-     x-        cx  \y  1  + 

As  this  is  an  evident  identity,  the  theorem  is  provinl. 

To  lind  the  condition  that  the  integrating  factor  may  be  a  function  of  x  only 
and  to  find  the  factor  when  the  condition  is  satisfied,  the  equation  (15)  which  fj. 
satisfies  may  be  put  in  the  more  compact  form  by  dividing  by  /x. 

,1/^^-JV^^=  — -—     or     J/^l^^_.V^i^!^  =  ^-^'_^:^.       (15') 
ndy  /J.  cx       cx        cy  cy  cx  cx        cy 

Now  if  /x  (and  hence  log  /j.)  is  a  function  of  x  alone,  the  first  term  vanishes  and 
d  log  fi     K,  -  K 


dx  N 


:/(x)     or     logM=   jf{x)dx. 


This  establishes  the  rule  of  type  lY  above  and  further  shows  that  in  no  other  case 
can  n  be  a  function  of  x  alone.   The  treatment  of  type  V  is  clearly  analogous. 

Integrate  the  equation  x*y  (3  ydx  +  2  xdy)  +  x'^  (4  ydx  +  3  xdy)  =  0.  This  is  of 
type  VII  ;  an  integrating  factor  of  the  form  fj.  =  xPy<^  will  be  assumed  and  the  ex- 
ponents p,  cr  will  be  determined  so  as  to  satisfy  the  condition  that  the  equation  be 
an  exact  differential.    Here 

P  =  fj.M  =SXP  +  4y<r  +  2  ^  4  j.p  +  -2,^,7  +1^  Q  _  ^_Y  =2XP  +  Sy^  +1  +  3  XP  +  3(/<^. 

Then  P^  =  3  (tr  +  2)  xP  +  ^y'^  +^  +  4{a  +  1)  xP  +  -•//«^ 

=  2  (p  +  5)  xP  +  ^y"  +1  +  3  (p  +  3)  xP  +  -'//-^  =  Q'^. . 

Hence  if  3  (cr  +  2)  =  2  (p  +  5)     and     4  (<r  +  1 )  =  3  (p  +  3), 

the  relation  P^  =  Q\.  will  hold.    'I'his  gives  a-  =  2,  p  =  1.    Hence  ^  =  xy~, 

and  r    (3  x-'y"^  +  4  rhf')  dx  +    C  Ody  =  I  x^y*  +  /■*//•"  =  C 

is  the  solution.  The  work  might  be  shortened  a  trifle  by  dividing  througli  in  the 
first  place  by  x^.  Moreover  the  integration  can  be  performed  at  sight  without  the 
use  of  (14). 

94.  Several  of  the  most  important  facts  relative  to  integrating  factors 
and  solutions  of  Mdx  -\-  Xdi/  =  0  will  now  l)e  stated  as  theorems  and 
the  proofs  Avill  be  indicated  below^ 

1.  If  an  integrating  factor  is  known,  the  corresponding  solution  may 
be  found ;  and  conversely  if  the  solution  is  known,  the  corresponding 
integrating  factor  may  be  found.  Hence  the  existence  of  either  implies 
the  existence  of  the  other. 

2.  If  F  =  C  and  G  =  C  are  two  solutions  of  the  e(|uation,  eithei'  must 
be  a  function  of  the  other,  as  (!  =  ^{F)  ;  and  any  function  of  eitliei'  is 


212  DIFFERENTIAL  EQUATIONS 

a  solution.  If  /u.  and  v  are  two  integrating  factors  of  the  equation,  the 
ratio  /i/v  is  either  constant  or  a  solution  of  the  equation ;  and  the  i)ro(l- 
uct  of  yu,  by  any  function  of  a  solution,  as  yu$(F),  is  an  integrating  fac- 
tor of  the  equation. 

3.  The  normal  derivative  dF jdn  of  a  solution  obtained  from  the 
factor  \i  is  the  product  yu.  Vj/-^  +  X'^-  (see  §  48). 

It  has  already  been  seen  that  if  an  integrathig  factor  /i  is  known,  tlie  corre- 
sponding solution  F  =  C  may  be  found  by  (14).  Now  if  the  stjlution  is  known,  tlie 
equation 

(IF  =  Fjlx  +  F'^dij  =  ii{Mdx  +  Ndy)     gives     F,'  =  /x.V,  F^  =  fj.N ■ 

and  hence  fx  may  be  found  from  either  of  tliese  ecjuations  as  the  quotient  of  a 
derivative  of  F  by  a  coefficient  of  the  differential  equation.  The  statement  1  is 
therefore  proved.  It  may  be  remarked  that  the  discussion  of  approximate  solutions 
to  differential  equations  (§§  86-88),  combined  with  the  theory  of  limits  (beyond  the 
scope  of  this  text),  affords  a  demonstration  that  any  ecjuation  Mdx  +  Xdij  =  0, 
where  M  and  N  satisfy  certain  restrictive  conditions,  has  a  solution  ;  and  lience  it 
may  be  inferred  that  such  an  equation  has  an  integrating  factor. 

If  fi  be  eliminated  from  the  relations  F^  =  fiM,  F'^  =  /jlN  found  above,  it  is  seen 
that 

MFy  -  NF^^  =  0,     and  similarly,     MG'^  -  .VG;  =  0,  (16) 

are  the  conditions  that  F  and  G  should  be  solutions  of  the  differential  ecjuation. 
Now  these  are  two  sinuiltaneous  homogeneous  ecjuations  of  the  first  degree  in  3/ 
and  N.    If  M  and  N  are  eliminated  from  them,  there  results  the  equation 


F'G'.-F'G'  =  0     or 


g:  g'a 


J{F,  G)  =  0,  (16') 


which  shows  (§  62)  that  F  and  G  are  functionally  related  as  refjuired.  To  show 
that  any  function  4'(F)  is  a  solution,  consider  the  eijuation 

3/*,;  -  x*;  =  (.VF,;  -nf'^.)  *'. 

As  F  is  asolution,  the  expression  3/^F,^—-VF'.  vanishes  by  (16),  and  hence  M^',—N^'^ 
also  vanishes,  and  <I>  is  a  solution  of  the  equation  as  is  desired.  The  first  half  of  2 
is  proved. 

Next,  if  IX  and  v  are  two  integrating  factors,  ecjuation  (15')  gives 

V  —  'ii^  —  V  L'i^  —  1/  ^  ' 'i"  "  _  V  ^  '"■^'  "     or     ¥  ^'^"-^/''  _  Y  L"!'-^/^  —  o 
cy  ex  cy  ex  cy  cx 

On  comparing  with  (16)  it  then  appears  that  log  (yu/f)  nmst  be  a  solution  of  tlie 
ecjuation  and  hence  /x/v  itself  nmst  be  a  solution.  The  inference.  ho\ve\fr.  would 
not  hold  if  fx/v  retluced  to  a  constant.  Finally  if  n  is  an  integrating  factor  leading 
to  the  solution  F  =  C,  then 

dF=  fx  {Mdx  +  Ndy),     and  hence     /x*  (F)  {Mdx  +  Xdy)  =  d  f*  ( F)  dF. 

It  therefore  appears  that  the  factor  pt<I>(F)  makes  the  e(|uation  an  exact  differen- 
tial and  must  be  an  integrating  factor.    Statement  2  is  therefore  wholly  pr(jved. 


COMMONER  ORDINARY  EQUATIONS  213 

The  third  proposition  is  proved  simply  by  differentiation  and  snbstitntion.    Por 

dF      cF  clx      cF  di/         ^^dx        ^^dy 

—  = 1 =  fj.M h  fxJy  — . 

dn       cx  dn       cy  dn  dn  dn 

And  if  T  denotes  the  inclination  of  the  curve  F  =  C,  it  follows  that 

dy          M            .           dy              N                                     dx              M 
tan  T  =  —  = ,         sm  t  =  —  = ,         —  cos  r  =  —  = . 

dx  N  dn      -y/M'^  +  N'^  o!'^      VM'^  +  N'^ 


Hence  dF/dn  =  /x  y/ M'^  +  N'^  and  the  proposition  is  proved. 

EXERCISES 

1.  Find  the  integrating  factor  by  inspection  and  integrate  : 

(or)  xdy  —  ydx  =  (x^  +  y^)  dx,  (j3)   (?/2  —  xy)  dx  +  x'^dy  —  0, 

(7)  ydx  —  xdy  +  logxdx  =  0,  {5)  y  {2  xy  +  e')  dx  —  ef'dy  —  0, 

(e)  (1  +  xy)ydx  +  (1  -  xy)xdy  =  0,  (f)  (x  —  y^)dx  +  '2xydy  =  0, 

(t;)  {xij^  +  y)  dx  -  xdy  =  0,     (6')  a  {xdy  +  2  ydx)  =  xydy, 

(i)  {x-  +  y-)  {xdx  +  ydy)  +  Vl  +  (x-  +  2/")  {ydx  —  xdy)  =  0, 

(k)  x"ydx  —  (x*  +  y^)  dy  =  0,  (X)  xdy  —  ydx  =  xVx^  —  y-dy. 

2.  Integrate  these  linear  equations  with  an  integrating  factor : 

(a)  y'  +  ay  =  sinte,  (/3)  y'  +  ycotx  =  secx, 

(7)  (X  +  1)  /  -2y  =  {x+  1)\  (5)  (1  +x2)  /  +  y  =  e ta"-'% 

and  (/5),  (5),  (f)  of  Ex.  4,  p.  200. 

3.  Show  that  the  expression  given  under  II,  p.  210,  is  an  integrating  factor  for 
the  Bernoulli  equation,  and  integrate  the  following  equations  by  that  method  : 

(a)  y'  —  y  tan  x  =  y^  sec  x,  (/3)  3  y'^y'  +  y^  -x  —  l, 

(7)  y'  +  y  cos  x  =  //"  sin  2  x,  (5)  dx  +  2  xydy  =  2  ax^y^dy, 

and  (a),  (7),  (e),  (rj)  of  Ex.  4,  p.  200. 

4.  Show  the  following  are  exact  differential  equations  and  integrate  : 

{a)  (3  x'^  +  6  xy-)  dx  +  (6  x'^y  +  4  y-)  dy  =  0,       (/3)  sin  x  cos  ydx  +  cos  x  sin  ijdy  =  0, 
(r)  (6x-2i/  +  l)dx  +  (2y-2x-3)di/  =  0,   (5)  (x^  +  3 x^'^) (Zx  +  (//^  +  3 x'^i/) cZy  =  0, 

2x?^+J.  ^^  _^  ?^::i^cZ2/  =  0,  (r)  (1  +  e'')  ^^-k  +  e'^  (1  -  -]  dy  =  0, 

(r;)  e-^'  (x-  +  2/'-^  +  2  x)  dx  +  2  yt^'dy  =  0,  (^)  {y  sin  x  —  1)  cZx  +  (y  —  cos  x)  dy  =  0. 

5.  Show  that  {Mx  —  Ny)~^  is  an  integrating  factor  for  type  III.  Determine 
the  integrating  factors  of  the  following  equations,  thus  render  them  exact,  and 
integrate : 

(a)  {y  +  x)dx  +  xdy  =  0,  (/3)  (?/2  —  xy)  dx  +  x^dy  =  0, 

(7)  (x'^  +  y'")  dx  —  2  xydy  =  0,  (5)  (x"?/'-^  +  xy)  ydx  +  (x^?/^  —  I) xdy  =  0, 

(e)   (V^-  1)  xdy  -  ( V^+  1)  ydx  =  0,       (f)  x^tZx  +  (3  xhj  +  2  y^)  dy  =  0, 

and  Exs.  3  and  0,  p.  206. 

6.  Show  that  the  factor  given  for  type  VI  is  right,  and  that  the  form  given  for 
type  VI]  is  right  if  k  satisfies  k  {qin  —  i^n)  =  q  {a  —  7)  —  i>  (/3  —  5). 


214  DIFFERENTIAL  EQUATIONS 

7.  Iiiteurate  tlie  following:  ('iiuations  of  types  IV-VII  : 

(a)   {y*  +  2ij)  dx  +  (j-r  +  2  y*  -As)dy  =  0,         (/3)   (x'^  +  i/  +  1)  dx  -  2  xydy  =  0, 
(7)   iSx^  +  ()Jcy  +  3y'')dx+{2y^  +  Sxy)dy  =  0,     (5)   {2x-^y' +  y)- {x^y  -  3x)  y'=0, 

( e)  (2  xhj  -  3  J/-*)  dx  +  (3  x^  +  2  xy'^)  dy  =  0, 

( f )  (2  -  '/)  sill  (3  X  -  2  y)  +  y'  sin  (x  -  2  y)  =  0. 

8.  By  virtue  of  prop(Jsiti(iii  2  above,  it  follows  that  if  an  equation  is  exact  and 
homogeneous,  or  exact  and  has  the  variables  separable,  or  homogeneous  and  under 
types  IV-VII,  so  that  two  different  integrating  factors  may  be  obtained,  the  solu- 
tion of  the  equation  may  be  oljtaiiied  without  integration.  Apply  this  to  finding 
the  solutions  of  Ex.  4  (/S),  (5),  (7)  ;  Ex.  5  {a),  (7). 

9.  Discuss  the  apparent  exceptions  to  the  rules  for  types  I,  III,  VII,  that  is, 
when  Mx  +  Xy  =  0  or  Mx  —  Xy  =  0  or  qrn  —  ]ni  =  0. 

10.  Consider  this  rule  for  integrating  Mdx  +  Xdy  =  0  when  the  equation  is  known 
to  be  exact :  Integrate  Mdx  regarding  y  as  constant,  differentiate  the  result  regard- 
ing y  as  variable,  and  subtract  from  X ;  then  integrate  the  difference  with  respect 
to  y.    In  symbols, 

C  =  f  {Mdx  +  Xdy)  =:   fMdx  +   f  ix  -  ^'-   f  Mdx\dy. 

Apply  this  instead  of  (14)  to  Ex.  4.  observe  that  in  no  case  should  either  this 
formula  or  (14)  be  applied  when  the  integral  is  obtainable  by  inspection. 

95.  Linear  equations  with  constant  coefficients.    The  type 

/•/"//  (l"~^i/  (1(1  ,  ,  ,^_^ 

"« 57'  + "'  Ta,^' +■■■  +  ■'-;&'  +  ""■'  =  -^  (■-)         (1  • ) 

of  differential  equation  of  the  n\\y  order  wliieli  is  of  tlie  first  degree  in 
?/  and  its  derivatives  is  called  a  Unnir  etjuation.  For  the  ]iresent  onh^ 
the  ease  where  the  eoetticicnts  a^^.  << ^.  ■■■,  "„_i,  "„  ai'e  constant  will  be 
treated,  and  for  eonveiiienee  it  will  he  assumed  tliat  the  etjuation  has 
heen  divided  through  hy  n.  so  tliat  tlie  coehicieiit  of  the  highest  deriva- 
tive is  1.  Then  if  differentiation  be  denoted  by  1),  the  equation  may  be 
written  si/t/J/oIirt/////  us 

(/>"  +  "i/>"  -'  +  ■■■  +  ".,  _,  1>  +  "„■)  //  =  A,  (17') 

where  the  symbol  J)  combined  with  constants  follows  many  of  the  laws 
of  ordinaiy  algebraic  ([uantities  (see  >;  70). 

The  simplest  e(|uation  would  be  of  the  first  oi'der.    Here 

'/'/  r 

-j a  1/ =  X      and      1/ =  r"^'    \  (■^"^'Xdx,  (18) 


as  may  be  seen  by  refei-enee  to  (11)  or  ((!).    Now  if  T>  —  a^  be  treated 
as  an  algebraic  symliol,  tlie  solution  may  be  indicated  as 

(/>__„^,,/=A-     an.l     .'/  =  --— -A,  (18') 


COMMOXEll   ORDINARY   EQUATION'S 


215 


where  the  operator  (Z>  —  ff^)'^  is  the  inverse  oi  D  —  a^ .  The  solution 
which  has  just  been  obtained  shows  that  the  interpretation  which  must 
be  assigned  to  the  inverse  operator  is 


D 


(*)  =  e"i^  j  e- "'''{*)  dx, 


(19) 


where  (*)  denotes  the  function  of  x  upon  which  it  operates.  That  the 
integrating  operator  is  the  inverse  of  I)  —  a^  may  be  ])roved  by  direct 
differentiation  (see  Ex.  7,  p.  152). 

This  operational  method  m-dx  at  once  be  extended  to  obtain  the  solu- 
tion of  equations  of  higher  order.    For  consider 


S  +  «i!7!  +  ''.^  =  -^'     o^"      iJy'  +  a^D  +  a.;)i/  =  X. 


dx- 


dx 


(20) 


Let  a^  and  «-,  be  the  roots  of  the  equation  1>-  +  <iJJ  -\-  a_^  =  0  so  that 
the  differential  equation  may  be  written  in  the  form 


\_1J-  —  (cfj  4-  a„)  D  +  a^fc.,]  //  =  A'     or      {D  —  ciTj)  (I)  —  <(_)  y  =  X. 
The  solution  ma}'  now  be  evaluated  l)y  a  succession  of  steps  as 


(20') 


1)  —  <i, 
V 


D  -  a. 


\'dj 


""=■'■  j  ^-("■-«2).'-      I  c- '''■'■  Xdx 


(20") 


The  solution  of  the  equation  is  thus  reduced  to  quadi'atures. 

The  extension  of  the  method  to  an  equation  (jf  any  order  is  immediate. 
The  first  ste})  in  tlie  solution  is  to  solve  the  c(|uation 

jy>  +  a/)"-'  +  ...  +  a,^_,l)-^>f,,  =  0 

SO  that  the  differential  e(|uation  niay  be  written  in  the  form 

( 7>  _  ,,-)  (/>  _  ,g  .  .  .  (/;  _  a,^  _0  (J)  -  a„)  //  =  X  ;  (17") 

whereu})On  the  solution  is  comprised  in  the  formula 


y 


=  e"n"-  Ce("" -'-'''»■'■  f  ■■■    rr("i-"^)'-  Cr-'>i'X(dxy,  (17'") 

Avhere  the  successive  integrations  are  to  be  performed  by  beginning 
upon  the  extreme  right  and  working  toward  the  left.  ^Foreover,  it 
a})pears  tliat  if  the  operators  1>  —  <i,,.  I)  —  '■(„_,.  ■  ■  ■ ,  I)  —  a\„  1>  —  <x^  were 
successively  applied  to  this  value  of  _//,  they  would  undo  the  work  here 


216  DIFFERENTIAL  EQUATIONS 

done  and  lead  back  to  the  original  equation.  As  n  integrations  are 
required,  there  will  occur  n  arbitrary  constants  of  integration  in  the 
answer  for  y. 

As  an  example  consider  the  equation  {Ifi  —  4:D)y  =  x".    Here  the  roots  of  the 
algebraic  equation  Ifi  —  i  D  =  0  are  0,  2,  —  2,  and  the  solution  for  y  is 

M  = X-  =    I  e'^^  I  e-'-'e-'^^  (  e--^x-{dx)''^. 

1JD--1D+  2  J         J  J  "■     ' 

The  successive  integrations  are  very  simple  by  means  of  a  table.    Then 
je^^x-dx  =  I  x^e-^  —  l  xe--^'  +  ^  e^^  +  Cj, 

fe-  ■*  •^  Ce-  ^x-{dx)-  =    f  ( '  x-e-  -  ^  —  J  xe-  ~-^- +  le---'  +  C\e-  *  =')  dx 

=  —  i  x-e-  - •'■■  —  J-  e- - -^^  +  C\e- *^+  C„ , 

?/  =    fe-^^-  fe-^''  Ce-^x~{dxY'  =   /"(—  i  x^  —  i  +  C'ie-2^+  C„e-^)dx 

=  —  yV  r''  —  I  ic  +  C',e-  2  a"  4-  C'„e-^  +  C'g. 

This  is  the  .solution.  It  may  be  noted  that  in  integrating  a  term  like  C■^e-*^  the 
result  may  be  written  as  C'je-*^,  for  the  reason  that  C\  is  arbitrary  anyhow  ;  and, 
moreover,  if  the  integration  had  introduced  any  terms  such  as  2  e---^,  J  e"^^,  5,  these 
could  be  combined  with  the  terms  C\e~-^,  C„e-"f,  (\  to  .simplify  the  form  of 
the  re.sults. 

In  case  the  roots  are  imaginary  the  procedure  is  the  same.    Consider 

Y  y  =  .sin  x     or     {!)'-  ■\-  l)y  =  .sin  x     or     (7J  +  z)  (Z»  —  i)  y  =  sin  x. 

dx" 

Then  ;/  = sin  x  =  e'-'"  |  e-  -  '•'"  /  e'^  sin  x  (dx)-,  i  =  V—  1 . 

The   formula  for   |  e«^' sin  hxdx,  as  given  in  the  tables,   is  not  applicable  when 

a-  +  b-  =  0,  as  is  the  case  here,  because  the  denominator  vanishes.  It  therefore  be- 
comes expedient  to  write  sin  x  in  terms  of  exponentials.   Then 

■    r     o/.  r  ;.<:''■''— 6" '•"/ 7  NO      r         ■           e'^— e-'> 
;/  =  e'-'      e-  -'■'      c'-' (dx)-  ;    tor    smx  = 

Now     — e'-'-  Ce--'-'-  j  {€-'■•  —  V)  (dx)-  =  -   .t''-  ft--'-'      -e-"'—  x  +  (\  \dx 

1  . r  1        1     , .       1     , .      , ,     , .      ^1 

2  1       I2i         2i  4  ^  -J 

X  e"  +  e-  '^       ,,  ^     . 

=  —  ^ ~ +  C\e-'^  +  a,e'-^-. 

Now  C\e-  '>  +  C',,e'>  =  {C,  +  C\)  ^"  "^  "'  "  +  {C„-  C^)  i  ^'^  "  ^"  '^ 


2i 

Hence  this  expression  may  be  written  as  C\  cosx  +  C.iSinx,  and  then 

y  =  —  I  X  cos  X  +  (^'i  cos  X  +  C^  sin  x. 

The  ,';olution  of  such  equations  as  these  gives  excellent  opportunity  to  cultivate  the 
art  of  manipulating  trigonometric  functions  through  exponentials  (§  74). 


COMMOXEK  OilDmARY  EQUATIONS  217 

96.  The  general  method  of  solution  given  above  may  be  considerably 
simplified  in  case  the  function  X  (x)  has  certain  special  fornis.  In  the 
first  place  suppose  A'  =  0,  and  let  the  equation  be  P  (D)  y  =  0,  where 
P  (D)  denotes  the  symbolic  polynomial  of  the  nth  degree  in  D.  Suppose 
the  roots  of  P{D)  =  0  are  a^,  a,^,  ■  ■  ■ ,  %.  and  their  respective  multiplicities 
are  vi^,  vi_^,  ■  ■  ■ ,  in^,  so  that 

{D  —  a-^,)'"i- ...  (D  —  ay"^(D  —  «j)""y  =  0 

is  the  form  of  the  differential  equation.    Now,  as  above,  if 

(D  -  a^y'ij  =  0,     then     ?/  =  — -~  0  =  e"'''    (  ■■■    J0(dx)"'\ 

Hence  v/  =  ,f^-(C^  +  f'^  -f  C^,--  +  •  •  •  +  C„,x'"^  -') 

is  annihilated  by  the  application  of  the  operator  (D  —  a^)'"',  and  there- 
fore by  the  application  of  the  whole  operator  P(D),  and  must  be  a  solu- 
tion of  the  equation.  As  the  factoi's  in  P(l))  may  be  written  so  tliat 
any  one  of  them,  as  (I)  —  a,)'"',  comes  last,  it  follows  that  to  each  factor 
(D  —  itiY'i  will  (correspond  a  solution 

y.  =  (f  >-((',,  +  C\^r  ^-  •  •  •  +  Cun-x>"i'^),  P (D)  11-  =  0, 

of  the  equation.    Moreover  the  sum  of  all  these  solutions, 

i  =  k 

y=Y^  ''"'"(f^'a  +  ^V  +  •  •  •  +  Ci,.,,-'i-^)^  (21) 

a  =  l 

will  be  a  solution  of  the  equation  ;  for  in  applving  PiJ^)  to  y, 
P  ij))  y  =  P  (D)  y^  +  /'  (P)  //,  +  ■••  +  P  (/))  //,  =  0. 

Hence  the  general  rule  may  be  stated  that:  The  solvflon  of  tlie  dif- 
ferential e(pt(itl<)n  P(Dyy  =  0  of  the  nih  order  ma y  lie  found  by  muJtiply- 
iiig  each  e"^  by  a  iMdynoinlal  (f(^i)i  —  Ij.s'/  der/rce  In  .r  {irhere  a  ii<  a  root  of 
the  equation  P  {D)  =  0  of  iindtipJielty  in  and  irliere  the  eoeficients.  of  tlie 
jxdynoinial  are  arbitrary)  and  addiny  tlie  rennlts.  Two  obstTvations 
may  be  made.  First,  the  solution  thus  found  contains  n  arl)itrary  con- 
stants and  niay  therefore  be  considered  as  the  general  s(jlution  ;  and 
second,  if  there  are  imaginary  roots  for  P  (/))  =  0,  tlie  exponentlah  ari.s- 
ing  from  tlie  pnre  imaginary  parts  of  tlie  roots  may  be  eonreiied  into 
t  rigono  met  rie  functions. 

As  an  example  take  (ZJ*  —  2  7/'  +  7)-)  //  =  0.  The  roots  are  1,  1,  0,  0.  Kence  the 
sohitiou  is  .,.,,,     ,    /,    ,   ,    /^,     ,    ^,    s 

Again  if  (7>*  +  4)  y  =  0,  the  roots  of  /;■*  +  4  =  0  are  ±  1  ±  i  and  the  solution  is 


218 


DIFFEREXTIAL  EQUATIONS 


or  y  =  e^  (O^e'^  +  C.e-  ")  +  e-  ^  (C,,e'''-  +  C^e-'^) 

=  e^'  (C\  cos  X  +  C'2  .sill  x)  +  e-  *'  (C3  cos  x  +  6'^  sin  x), 

where  the  new  Cs  are  not  identical  witli  the  old  6"s.    Another  f(n-m  is 

y  =  e^  A  cos  (x  +  7)  +  e-^  iJ  cos  (x  +  5), 

where  7  and  5,  A  and  7),  are  arbitrary  constants.    For 


Cj  cos  X  +  Co  sin  x  =  Vt'f  +  C'.| 


cosx  + 


and  if     7  =  tan  -  1  / '^\,     then     C^  cos  x  +  C.,  sin  x  =  Vc'^-  -f  C'^-  cos  (x  + 


7). 


Xcxt  if  A'  is  not  zero  but  if  any  one  solution  I  can  he  foiaid  so  that 
]^(^1)')  I  ^  X,  ilnni  a  solution  contalnlurj  n  arliltra rij  (■(mstants  'iiuiij  be 
found  hij  adding  to  I  tJte  sitlatlon  of  l\I))i/  ^=  0.    Eur  if 

/>(/>)  /  =  X     and     r{D)  y  =  0,     then     P(/))  (f  -(-  7/)  =  A'. 

It  therefore  remains  to  devise  means  for  finding  on(>  solution  /.  This 
solution  I  ]nay  he  found  hy  the  long  method  of  (17'"),  where  the  inte- 
gration may  he  shortened  In'  omitting  the  constants  of  integration  since 
only  one,  and  not  the  general,  ^■allle  of  the  solution  is  needed.  In  the 
most  important  cases  which  arise  in  practicte  there  are,  liowe\'er,  some 
very  short  cuts  to  the  solution  /.  The  solution  /  of  7'(/>)//  =  A  is 
called  the  pa rtlcular  bifrgral  of  the  equation  and  tlie  general  solu- 
tion of  l\l))i/  =  0  is  called  the  (■oiiqJeiiD'ntari/  function  for  the  e(|ua- 
tion  r{D)  //  =  A. 

Suppose  that  A'  Is  a  jxd i/nonilol  In  .>'.  Solve  symbolically,  arrange 
I'(D)  in  as(!ending  ])owers  of  J),  and  divide  oui;  to  ])(nv('rs  of  7>  e(|ual  to 
the  order  of  tlie  polynomial  A.    Then 


p(i))r=x, 


I  = ■  A  = 


A, 


(22) 


where  the  I'emainder  H{1>)  is  of  hlfjhci'  oi'dcr  in  I)  tlian  A'  in  x.    Then 

7> (7))  /  =  P (79)  qijt)  X  +  7.'  (7>)  A,  R  {If)  X  =  0. 

Hence  Q(7;).r  may  l)e  taken  as  /,  since  J' (D)  Q(D)  X  =  J>(I))  f  =  X.  By 
tliis  method  the  solution  /  may  be  found,  wlicn  X  is  a  polynomial,  ^r.s- 
'/■(ijildl;/  OS  l'(l>)  can  he  dlcldcd  Into  1  ;  tli(^  solution  of  J'(/))  //  =  0  may 
1h'  written  down  by  (21);  and  tlie  smn  of  /and  tliis  will  be  the  recpiired 
soluticju  ol'  7'(  1>)  1/  =  A  containing  ?i  constants. 

As  an  example  considci'  {If-  +  4  //-  +  S  JJ)  1/  =  x~.    The  work  is  as  follows : 


3IJ  +  -i  D-  +  1)^ 


] 


1 


J)-6-\-  iD  +  D-^' 


Vd     'J 


id7  l\J))j 


COMMONER  ORDINARY  EQUATIONS  219 

Hence  I  =  (2(D)x^  =  ^  (- -- D  +  - 1 A  x^  =  -x^-^x^  +  ~  x. 

For  D^  +  4D^  +  SD  =  0  the  roots  are  0,-1,-3  and  the  complementary  function 
or  solution  of  P{D)y  =  0  would  be  C^  +  C^e~^  +  C^e-^^'.  Hence  the  solution  of 
the  equation  P  {D)  y  =  xr  is 

2/  =  6\  +  G„e-^-  4-  C^e-^^  +  \x^  —  f,  x^  +  |«  x. 

It  should  be  noted  that  in  this  example  Z)  is  a  factor  of  P{D)  and  has  been  taken  out 
before  dividing  ;  this  shortens  the  work.  Furthermore  note  that,  in  interpreting 
1/D  as  integration,  the  constant  may  be  omitted  because  any  one  value  of  I  will  do. 


and         P  (D)  e"""  =  P  (a)  e""" ;     hence     P(D) 


Ce" 


But  P{D)  I  =  Ce%     and  hence     I  =  -j^  e"""  (23) 


97.  Next  suppose  that  X  =  Cc"^.    Now  De"''  =  ae"'',  Dh"''  =  «^e''% 

^    C 
LP  (a) 
C_ 

'(a) 

is  clearly  a  solution  of  the  eqxiation,  provided  a  is  not  a  root  of  P(D)  =  0. 
If  P  (a)  =  0,  the  division  by  P  (a)  is  inipossil)le  and  tlie  quest  for  /  has 
to  be  directed  more  carefully.  Let  «  be  a  root  of  multiplicity  vi  so  that 
P(D)  =  (D  —  ay"P^(D).    Then 

C 

Pj(P)  (D  -  a)'"!  =  Ce""^,  (D  -  a)">I  =  — —  e"'', 

and  I  =  -— -  e""-  \  \   (ih-y^  =       \    '  -  •  (23') 

For  in  the  integration  the  constants  may  be  omitted.  It  follows  that 
when  X  =  Ce"'',  the  solution  1  may  be  found  hi/  direct  siihufltufion. 

Now  if  X  broke  up  into  the  sum  of  terms  A'  =  A'^  +  A',  +  •  •  •  and  if 
solutions  I^,  /.„  •  •  •  were  determined  for  each  of  the  equations  P{D)I^=  X^, 
P  (D)  I^  =  A.„  •  •  • ,  the  solution  /  corresjionding  to  A'  Avould  be  the  sum 

7j  +  /.,  4 .    Thus  it  is  seen  that  the  above  short  methods  apply  to 

e(}uations  in  which  A  is  a  sum  of  terms  of  the  form  Ca-'"  or  Ce'^''. 

As  an  example  consider  (D*  —  2  7/-  +!)?/  =  e^.  The  roots  are  1,  1,  —  1,  —  1, 
and  a  =  1.    Hence  the  solution  for  I  is  written  as 

(D+l)2(D-l)2I  =  e%         (i)-l)2I  =  ie^,        J^ie-^x^. 

Then  y  =  e^{G^  +  C„x)  +  e-^-{C.^  +  C^x)  +  i  e-^-x-. 

Again  consider  (D^  _52)^.0)y  —  x  +  e""'.  To  find  the  Ij  corresponding  to  x, 
divide.  ^  5 

+  — 7)  +  .-.  hc  =  -x  + 


'      6-  57)  +  7)2         \6      36  /         6         36 

To  find  the  I^  corresponding  to  e'"-"",  substitute.    There  are  three  cases, 

7.,  = e '»•'■,        7.,  =  xe^^-,         7„  =  —  xc"--'-, 

m-  —  5  ??i  +  6 


220  DIFFEREXTIAL   EQUATIOXS 

according  as  m  is  neither  2  nor  3,  or  is  3,  or  is  2.    Hence  for  the  complete  solution, 
y  =  C\e^'-  +  C-.f^-'-  +  --X  +  ^  + 


6         36      m-  —  5  ni  +  6 
when  m  is  ncitlier  2  nor  3  ;  Viut  in  these  special  cases  the  results  are 

y  =  Cie^-^  +  C.^c-^^--  +  1  X  +  3\  -  xe^^,         y  =  C^e^^  +  Cjfi^  +  -^  x  +  /..  +  xe^^. 

The  next  case  to  consider  is  where  X  is  of  the  form  cos  ^x  or  sin  ^x. 

If  these  trigonometric;  functions  he  expressed  in  terms  of  exponentials, 

the  solution  may  he  conducted  by  the  method  above ;  and  this  is  per- 

luips  the  best  method  when  ±  ^/'  are  roots  of  the  equation  /'(/))  =  0. 

It  may  b(^  noted  that  this  method  would  apply  also  to  the  case  where 

A'  might  l)e  of  the  form  ('"'cos  (ix  or  r"''  sin  fix.    Instead  of  splitting  the 

trigonometric  functions  into  two  ex])onentials,  it  is  possible  to  combine 

two  trigonometric  functions  into  an  exponential.    Thus,  consider  the 

equations 

7'  (Z)j  y  =  r"'-  cos  ^x,  P(D)  y  =  r"''  sin  (3x, 

and  P  (D)  ij  =  (-"■■  (cos  jix  +  I  sin  (ix)  =  r("  +  ^'>'\  (24) 

The  solution  /  of  this  last  equation  may  be  fcnmd  and  split  into  its 
real  and  imaginary  ])arts,  of  Avhich  the  real  })art  is  the  solution  of  the 
equation  involving  the  cosine,  and  the  imaginary  part  the  sine. 

When  A  has  the  form  cos  /3x  or  sin  fix  and  ±  (31  are  not  roots  of  the 
equation  P(D)  =  0,  there  is  a  very  short  method  of  finding  /.    For 

n-  cos  f3x  =  —  /?'-  cos  (3x     and     7)-  sin  ^x  =  —  fS'-  sin  fSx. 

Hence  if  P(J>)  bt;  written  as  /'///-)  -f  DPjT)-)  hy  collecting  the  even 
terms  and  the  odd  tei'uis  so  that  /'^  and  7',,  are  both  even  in  D,  the 
solution  may  be  carried  out  syml)olically  as 

11  1 


P{D)      ■  ■         P^ilf-)  +  DI'SJ)-)      '  "        ]\{-  /3-)  +  DPI-  P-) 
pi-p'-)-]>l>,^(-(S') 


cos  .r, 


^^  ^  —  cos  X.  ('25^ 

By  this  device  of  substitution  and  of  rationalization  as  if  D  were  a  sui-d, 
the  differentiation  is  li-insfci'i-cd  to  tlie  numerator  and  can  be  ])crformt'd. 
This  method  of  ])roc('(lure  may  be  justiticd  directly,  or  it  may  be  made 
to  de})end  iqion  tliat  of  the  jiaragrajih  above. 

(>)nsi(lcr  tlie  example  (//-  +  1)//  :=  cos.c.  Here  /3/  =  /  is  a  root  of  7>-  +  1  =  0. 
As  an  oiJcratnr  7/-  is  e(|nivalent  to  —  1.  and  the  rationalization  method  will  not 
work.    If  the  tirst  solution  l)e  follewed,  the  method  of  solution  is 

1        C''>  1        c-  '■'■  1       (■'■'■  1       r-  '■'■        1    ,      .  .  ,       1       . 

H ;; " :  — ;  —    :  - -—  =  -  -  [•'^f-''''  —  xc~  '■']  =  r  X  sui  x. 


//-  +12        7/-  +  1     2  ]>-  Hi      ])  +  I   -ii        4 1  2 

If  the  secontl  suui^'estion  he  followed,  the  solution  may  be  found  as  f 


COMMONER  ORDIXARY   EQUATIONS  221 

1                xe'^ 
(D-  +  1)  /  =  cos  X  +  i  sin  x  =  e'^,         /  = e'-^  = 

T       a;  ,  .   .      ,       1      .  1  . 

Now  I  =  —  (cos X  +  I  sin  X)  =  -  X  sin  X ix  cos x. 

2 1  ^2  2 

Hence  /  =  ^  x  sin  x         for     {D^  +  1)  /  =  cos  x, 

and  I=  — Jxcosx     for     (D- +  1)  /  =  sinx. 

The  complete  solution  is  ?/  =  C\  cos  x  +  Co  sin  x  +  J  x  sin  x, 

and  for  {D^  -\-\)y  =  sin  x,  y  =  C,  cos x  +  C„  sin  x  —  i  x  cos x. 

As  another  example  take  {!)-  —  3  D  +  2)  ij  =  cosx.    The  roots  are  1,  2,  neither 
is  equal  to  ±  ^i  =  ±  i,  and  the  method  of  rationalization  is  practicable.    Then 

1                                1                     1  +  3D  1   ,  „   .      ^ 

cosx= cosx  =  —  (cosx  —  3sinx). 


lJ-^--SD  +  2  1-3Z>  10  10 

The  complete  solution  is  y  =  C\e-''+  C.^c--^  +  ^'^(cosx—  Ssinx).    The  extreme 
simplicity  of  this  substitution-rationalization  method  is  noteworthy. 

EXERCISES 

1.  By  the  general  method  solve  the  equations  : 

dx^         dx  dx^         dx-         dx 

(7 

iv 


1)2  _  4  i)  +  2)  ?/  =  X.  (5)  {/>''  +  Ifi  -  in-  4)  y  =  X, 

Iy^  +  5  D2  +  0  jj)  y  =  j^  (f )  (7J-2  +  /;  +  1 )  y  =  je  .^ 

Z>2  +  i,  +  1)  y  =  sin  2  X.  (^)  (7)2  _  4)  (/  =  X  +  C^  ^ 

7/2  4-  3  i»  4-  2)  ?/  =  X  +  cos  x.  (k)   (7>«  -  4  i*-)  y  =  1  -  sin  x, 

7^2  + i)y  =  cosx,              (m)  (7/2  +  l)y  =  secx,             {v)  {D- -{■  \)y  =  im\x. 

By  the  rule  write  the  solutions  of  these  eijuations  : 

j/2  +  o  2,  ^  2)  y  =  0,  (P)   {Ifi  +  3  Ifi  +  TJ-  5)  y  =  0, 

D-  l)"y  =  0,  (5)  (/>  +  27/-2  +  1)//  =  0, 

J>i  _  3  Lr2  +  i)y  =  0,  (f )  (D*  -  If'  -  U  IJ-  -\\J)-4)y  =  0, 

Z>3  _  0  i»2  4.  9  7;)  y  =  0,  (&)   {ly^  -  4  7;-  +  8  D-  -  8  7;  +  4)  y  =  0, 

7>5  _  2  7>  +  7>':)  y  =  0,  (k)   {If'  -  I)-  +  /^)  .V  =  0, 

7)4  _  1)2^  ^  0,  (m)  (7/5  -  137>5  +  2nifi  +  82  J)  +  104)?/  =  0. 

By  the  short  method  solve  (7),  (5),  (e)  of  Ex.  1,  and  also  : 

l)i  -\)y  =  x\  (p)  (7>'5  -  6  IP  4-117)-  (',) }/  =  X, 

7>'5  4-  3  7)2  4-  2  I))  y  =  x2,  (5)  {Jfi  -  3  7/2  -  G  7)  4-  8)  y  =  x, 

7>'5  4-  8)  ?/  =  X*  4-  2  X  4-  1,  (r)  (7>''  -  3  7^2  _  7;  4-  3)  //  =  x-. 

7>4  -  2  7>' 4- 7)2)  7/ =  X,  (0)  (7/*4-27>''4-37/24-27)4-l)y  =  l  +  x4-x2, 

7>'5  -  1)  2/  =  x2,  (0  (7/4  -2  7/^  +  7/2)  y  =  x^ 

By  the  short  method  solve  (a),  (^).  (<9)  of  Ex.  1,  and  also  : 

7)2  _  3  7)  4.  2)  2/  =  e-^,  (/3)   (7>»  _  7/^  _  3  7/2  4-  5  7)  -  2)  y  =  c^-^ 

7)2  -  2  7)  4-  1)  ?/  =  e^  (5)  (T*-''  -  3  7»2  4-  4)  y  =  e^-^, 

7)2  +  1)  2/  =  2  e-^  4-  x^'  -  X,  (f)  (7>^  4-  i)  y  =  3  +  e-^'  4-  0  ^2.--, 

7)4  4.  2  7)2  +  1)  ?/  =  e-^  4-  4,  (^)   (7>^  4.  3  7)2  4-  3  7>  4-  1)  2/  =  2  e-^, 

7/2  _  2  7/)  y  =  e2.-  4-  1 .  (k)  (7)3  4-  2  7)2  +  7/)  y  =  c2.'-  4-  x2  +  x, 

7»2  _  a^)  y  =  c«-'-  4-  e'---,  (^)   (7)2  -  2  a7;  4-  «2)  y  =  c'^  4-  1. 


222  DIFFERENTIAL   EQUATIONS 

5.  Solve  by  the  short  method  (77),  (t),  (k)  of  Ex.  1,  and  also  : 

(a)  (Z»2  -  Z)  -  2)  y  =  sin  x,  (/3)  (Z>2  +  2  D  +  1)  2/  =  3  e2^  -  cos  j, 

(7)  (D'-i  +  4) 2/  =  x2  +  cos X,  (5)  (i*^  +  -Z^^  -  i*  -  1)  2/  =  cos  2  x, 

(e)  (D2  +  1)^2/  =  cosx,  (f)  {U^  -  Ifi+  I)-l)y  =  cosx, 

( r?)  (2)2  -  5  Z)  +  6)  2/  =  cos X  -  e2^,        ((9)  (Z>3  -  2  i»2  _  3  i*)  ?/  =  3  x"-  +  sin  x, 

( 1 )  (Z»2  _  i)2y  =  .sin  X,  (/c)  (2)2  +  3  i;  +  2)  ?/  =  e2^  sin  x. 

(X)  (7>  -  1)2/  =  e-^cosx,  (/x)  (i/''  -  3Xi2  +  4i»-  2)  y  =  e---+  cosx, 

{v)  (Z»2  —  2 1>  +  4) y  =  e^ sin x,  (0)  (1/2  +  4)  1/  =  sin 3 x  +  c-'-  +  x2, 

"_       J*  V  3 
(tt)  (D"  4-  1)  ?/  =  sin  I  X  sin  1  x,  (p)   (/>'  +  1)  ?/  =  e2a^  sin  x  +  e2  sin  — — , 

( 0-)  (Z>2  +  4)  2/  =  sin2  X,  (t)   (Z)4  ^  32  x>  +  48)  y  =  xc-  2 ■'■  +  e^ x  cos  2t  x. 

6.  If  X  has  the  form  e-^^A", ,  show  that  I  = e"-'X,  —  e^'- X, . 

^  P{I))  ^  F{D+a)      ' 

This  enables  the  solution  of  equations  %Yhere  X^  is  a  polynomial  to  be  oljtained  Ijy 
a  short  method  ;  it  also  gives  a  way  of  treating  equations  where  A'  is  (fi-''  cos/3x  or 
e^^  sin /3x,  but  is  not  an  improvement  on  (24)  ;  finally,  combined  witli  the  second 
suggestion  of  (24),  it  covers  the  case  where  A' is  the  product  of  a  sine  or  cosine  by 
a  polynomial.  Solve  by  this  method,  or  partly  by  this  method,  (f)  of  Ex.  1  ;  (k),  (X), 
(y),  (p),  (t)  of  Ex.  5  ;  and  also 

(a)  (D2  _  2  D  +  1)  ?/  =  x2e3^,  (jS)  {I/^  +  3  1)2  +  3  7)  +  1)  2/  =  (2  -  x2)  e-^, 

(7)  (D2  4.  7i2)  y  =  x*e^,  (5)  (Jj^  _  9  J>"  _  3  Jfi  +  4  7;  +  4)  ^  ^^  j-^gx^ 

(e)  {Ifi  -lD-6)>j  =  e2^(l  4-  X),  it)  {D-ly^y  =  e'-  +  cosx  +  x-c^, 

(17)  (D  -  Ify  =x-  x^e^,  {6)  (//-  +  2)  ?/  =  x2e«^  4-  e-^  cos  2  x, 

(t)  (!>'  —  1)7/  =  xe^  4-  cos2x,  (k)  (Z»2  _  i)y  =  .^  sin  x  4-  (1  4-  x2)  e-'', 

(X)  (D2  4-  4) ?/  n=  X  sin  x,  {ij.)  {Ij^  -]-  21)-  +  \)y  -  x-  cos  ax._ 

( v)  (2)2  4.  4)  ,y  =  (j  sin  x)2,  (0 )  (i>2  _  2  Jj  4-  4)2?/  =  xe-^  cos  V3  x. 

7.  Show  that  the  substitution  x  =  e',  Ex.  0.  p.  152,  changes  equations  of  the  type 

x^lJ"y  4-  r^jX"  -li^"  - 1//  4-  •  •  •  +  «„  -  vi-Dii  4-  a„y  =  X  ix)  (20) 

into  equations  with  constant  coefficients  :  also  that  a^r  +  b  =  e'  would  make  a  simi- 
lar simplification  for  equations  whose  coefficients  were  powers  of  ax  +  b.  Hence 
integrate : 

(a)   (x2ii2  _  J.JJ  ^  2)  y  =  X  log  x,  (/3)   {x^If^  -  x2X»2  4-  2  xl)  -  2)  y  =  x^  4-  3  x, 

(7)   [(2x-l)5/>H(2x-l)Z»-2]?/  =  0,       (5)   (x--J>2  4-3x/>'4- ]),'/ =  (1 -.'•)-2. 
(e)  {x^iy^  +  xD  —  1)  y  =  X  log  X,  (f)   [(x  4-  ly-lJ-  —  4  (x  4-  1  )/>  4-  (i]  ?/  =  x. 

(ij)   (x2i»2  4.  4  jX»  4-  2) 2/  =  e^,  (^)  (x^l/2-3x27_;4-x)//=  logxsinlogx  4-1. 

( t)   (x*7>'  4-  Ox^/>'  4-  4  x-D"  -  2  xD  -  i)  ij  =  x-  +  2  cos  log  x. 

8.  If  L  be  self-induction.  I!  resistance,  C  capacity,  i  current,  q  charge  upon  the 
plates  of  a  condenser,  and /(f)  the  electromotive  force,  then  the  differential  equa- 
tions for  the  circuit  are 

<">  'f  +  7 1'  +  /r  =  }  •«'>■       <«  ?'  +  f  7;  +  tV^  =  7  /■«• 
at-      L  at      L  t       L  at-      L  dt      L  C      L 

Solve  (a)  when/(i)  =  e-  "'  sin  bt  and  (^)  when/(<)  =  sin  bt.  Keduce  the  trigonometric 
part  of  the  particular  solution  to  the  form  K  sin  {bt  4-  7).  Sliow  that  if  R  is  small 
and  h  is  r.carly  equal  to  1  /a  LC.  t!ie  amplitude  K  is  large. 


COMMONER  ORDIXARY  EQUATIONS  223 

98.  Simultaneous  linear  equations  with  constant  coefficients.    If 

there  be  given  two  (or  in  general  n)  linear  equations  with  constant 
coefficients  in  two  (or  in  general  n)  dependent  variables  and  one  inde- 
pendent variable  t,  the  symbolic  method  of  solution  may  still  be  used 
to  advantage.    Let  the  ecjuations  be 

{ajr  +  .-•,  /)"  -'  +  ■■■+  "„)  :r  +  0>,p"'  -f  h^'"  -'  +  ■■■+  h^)  y  =  R  (f), 
{c^D"  +  r^D"  -1  +  •  •  •  +  .■ ,)  .7-  +  Q/^D"  4-  'fjy-'  +  •  •  •  +  (/,)  !/  =  S  (t),  ^^ '  ^ 

when  there  are  two  variables  and  where  D  denotes  differentiation  by  t. 

The  equations  nuiy  also  be  Avritten  more  briefly  as 

J\(D)  .-/■  +  Q^(D)  tl  =  n     and     PjJ))  x  +  Q.,(D)  j/  =  S. 

The  ordinary  algebraic-  process  of  solution  foi-  x  and  //  may  l)e  employed 
because  it  depends  only  on  such  laws  as  are  satisfied  equally  by  the 
symbols  D,  1\(1>),  Q^0>),  and  so  on. 

Hence  the  solution  for  ./■  and  y  is  found  by  multiplying  by  the  ap- 
propriate coefficients  and  adding  the  e(|uations. 


l\(D).r  +  q^(D)i/  =  R, 

PjD),r  +  QJI))i/  =  S. 


Then  II\(D)  QJD)  -  PjD)  Q^(l))^  j-  =  Qjl))  11  -  (1^(1  >)  S, 

It  Avill  be  noticed  that  the  coefficients  by  which  the  eriuations  are  multi- 
plied (written  on  the  left)  are  so  cliosen  as  to  iiuikc  tlie  coefficients  of 
X  and  1/  in  the  solved  form  tlie  same  in  sign  as  in  other  respects.  It  may 
also  be  noted  that  the  order  of  P  and  (t  in  the  symbolic  ]iroducts  is  im- 
material. IJy  ex]>an(ling  tlie  operator  P^{D)  Qo(J^)  —  J' J ^l)  '^lij^)  '^"^  <'t'i'taiii 
polynomial  in  ]>  is  o1»tained  and  liy  a])plying  the  o])erators  to  A'  and  >' 
as  indicated  certain  fund  ions  of  t  are  obtained.  Each  eqi;ation,  whether 
in  ./•  or  in  //,  is  quite  of  the  form  that  has  l)een  treated  in  §§  95-97. 

As  an  example  consider  tlic  solution  for  x  and  ij  in  the  case  of 

2^-  ^'^-  4.C  =  2^         2'i'':  +  i'^-  .3  V  =  0; 
dC^       dt  dt  dt 

or  (2  I)-  -  i)j-  -  ])!/  =  2t,  2  Jlr  +  (4  7;-  ?,) ;/  =  0. 

Solve  4  7;  -  .3  I    -  2  7;   |  (2  7;-  -  4)  r  -  Ih/  =  2  t 

1)  2  7)-  -  4 :         2  ])f  4-  (4  7;  -  .8)  y  =  0. 

Then  [(4  7>  -  .3)  (2  7/^  -  4) -1- 2  7/-^]  x  =  (4  7>  -  .S)2<, 

[2  7)2 -f  (2  7J-^  -  4)  (4  7)  -  .3)]  y  =  -  (2  7))  2  i, 

or        4  (2  Ifi  -  If--  ^  T)  +  .3)  .r  =  8  -  (3 1,         4  (2  If'  -  D- -  i  IJ  +  ?j)y  =  -  4. 

The  roots  of  the  polynomial  in  D  are  1,  1.  —  H  ;  and  the  particular  solution  Ij^  for 
J  is  —  ^  f.  and  I,,  for  ;/  is  —  J.    Hence  the  solutions  have  the  form 

.r  =  ( /',  +  CJ)  <:'  -f  r,(-  i '  -  i  ^         y  =  (^1  +  K./)  c'  +  K,/r ' '  -  1  • 


224  DIFFERENTIAL  EQUATIONS 

The  ai'bitraiy  constants  wliicli  are  introdnced  into  the  solutions  for  a; 
and  II  are  not  independent  nor  are  they  identical.  The  solutions  must 
be  substituted  into  one  of  the  ciiudtions  to  establish  the  necessory  relations 
between  the  eonstants.  It  will  be  noticed  that  in  general  the  order  of  the 
equation  in  1)  for  x  and  for  ij  is  the  sum  of  the  orders  of  the  highest 
derivatives  which  occur  in  the  two  equations,  —  in  this  case,  3  =  2  4-1- 
The  ordt'r  may  l)e  diminislied  h\  cancellations  which  occur  in  the  formal 
algebraic  solutions  for  x  and  y.  In  fact  it  is  conceivable  that  the  coeffi- 
cient Pj(?.,  —J'M^  of  X  and  y  in  the  solved  equations  should  vanish  and 
the  solution  become  illusory.  Tliis  case  is  of  so  little  consequence  in 
practice  that  it  may  be  dismissed  with  the  statement  that  the  solution 
is  then  either  impossible  or  iudcttM-minate ;  that  is,  either  there  are  no 
functions  ./■  and  //  of  ;"  which  satisly  the  two  given  differential  equations, 
or  there  are  an  intinite  number  in  each  of  which  other  things  than  the 
constants  of  integration  are  arbitrary. 

To  finish  tlie  example  above  and  determine  one  set  of  arbitrary  constants  in 
terms  of  the  other,  substitute  in  the  second  differential  eijuation.    Then 

2  (CjC'  +  Coe«  +  CJe'  -  |  C^e"  ^' -  l)  +  -i  (K^c>  +  K.,e>  +  KJC  -  3  K^c'  i  ') 

-  3  (h\e>  +  K./e>  +  K^/'  i '  -  i)  =  0, 
or  e'(2  C\  +  2  T.  +  A',  +  Jf.,)  +  tc<{2  (\_  +  Jv'.,)  -  3  e" i  \C\  +  3  iv,)  =  0. 

As  the  terms  e',  tc',  e~  ^ '  are  independent,  the  linear  relation  between  them  can 
hold  only  if  each  of  the  coefficients  vanishes.    Hence 

C3  +  3  iig  =  0,         2  C„  +  A'._,  =  0,         2  C\  +  2  C'._,  +  A',  +  A'.,  =  0, 

and  C'3  =:  -  3  A'3 ,         2  ( ',  =  -  A'., ,         2  C\  =  -  7v , . 

Hence  x  =  (C,  +  CJ) c>  -  3  K.e~  2 '  -  H,         y  =  -  2  (r ',  +  CJ)  t>  +  K^e'  i' -  \ 

are  the  finished  solutions,  where  C\.  ('.,.  A',,  are  three  arbitrary  constants  of  inte- 
gration and  nii^ht  e(jually  well  be  denoted  liy  C\.  ('., .  f'g,  or  7vj.  K.^.  7\',,. 

99.  One  of  the  most  important  applications  of  the  theory  of  sinudtaneous  equa- 
tions with  con.'itant  coefficients  is  to  tlie  theory  of  sniuU  vihrationn  iibout  a  state  of 
equilibrium  in  a  ronservative*  dynamicnl  sy-stem.  If  q^ .  7., .  •  •  • .  r/„  are  n  coordinates 
(see  Exs.  19-20,  p.  112)  which  specify  the  po.sition  of  the  system  measured  relatively 

*  The  potential  cncr^iy  V  is  defined  as  —  dV=  fl  \V  =  Qi<!'j^  +  0-2'''y2  -\ +  Qndqn, 

<^'ii         (Qi         i'li  ('li         f'/i-         <^qi 

This  is  tjic  iiiimcdiatf  extension  of  Q^  as  ^iveii  in  Ex.  l'.i,  ji.  112.  Here  '/IT  denotes  the 
differential  of  work  ami  nw  =  2F,-/r,  =.  2(A>/.^+  !>///,•  -^  Z,'/2,).  To  tind  Q,-  it  is 
generally  quickest  to  eonipute  '/IF from  this  relation  with  '/.r,-,  Jv,-.  dz,  pxpresse<l  in  terms 
of  the  differentials  '/'/i,  •  •  •  ,  ''7,,.  Tlie  generalized  forces  Qi  are  then  the  coetHcients  of 
ilqi.  If  there  is  to  be  a  potential  V.  the  ditTerential  'Hrmust  be  exact.  It  is  frequently 
easy  to  find  I'  directly  in  terms  of  q■^,  ■■■.  q„  rather  than  through  the  mediation  of 
Oi .  •  ■  ■  .  (Jn'-  wlien  this  is  not  so,  it  is  usually  better  to  leave  the  equations  iu  the  form 

<l  cT       cT  .         , 

--  .. .—  =  Qi  rather  than  to  mtrodueu  J'and  L. 

dt  Cq,         Cqi 


COMMOI^ER  ORDIXARY  EQUATIONS  225 

to  a  position  of  stable  equilibrium  in  which  all  the  q'a  vanish,  the  development  of 
the  potential  energy  by  Maclaurin's  Formula  gives 

y{(li ,  9-2 ,  •  •  ■ ,  Qn)  =  Vo  +  ^lili  >  ^2 '  •  •  • '  f/«)  +  ^-li^h ,  ^2 ,  •  •  • ,  9«)  +  •  •  • , 

where  the  first  term  is  constant,  the  second  is  linear,  and  the  third  is  quadratic,  and 
where  the  supposition  that  the  q's  take  on  only  small  values,  owing  to  the  restriction 
to  small  vibrations,  shows  that  each  term  is  infinitesimal  with  respect  to  the  preced- 
ing. Now  the  constant  term  may  be  neglected  in  any  expression  of  potential  energy. 
As  the  position  when  all  the  q's  are  0  is  assumed  to  be  one  of  equilibrium,  the  forces 

cV  ^,  cV  ^  cV 

Qi  =  -— '        Q2  =  -  —  >     •■•'    Qn  =  -  — 

cq-i  cq^  cq„ 

must  all  vanish  when  the  ^'s  are  0.  This  shows  that  the  coefficients,  {dV/dq,)o  =  0, 
of  the  linear  expression  are  all  zero.  Hence  the  first  term  in  the  expansion  is  the 
quadratic  term,  and  relative  to  it  the  higher  terms  may  be  disregarded.  As  the 
position  of  equilibrium  is  stable,  the  system  will  tend  to  return  to  the  position 
where  all  the  ^'s  are  0  when  it  is  slightly  displaced  from  that  position.  It  follows 
that  the  quadratic  expression  must  be  definitely  positive. 

The  kinetic  energy  is  always  a  quadratic  function  of  the  velocities  qi,  q,,,-  •  ■ ,  ('u 
with  coefficients  which  may  be  functions  of  the  r^'s.  If  each  coefficient  be  expanded 
by  the  Maclaurin  Formula  and  only  the  first  or  constant  term  be  retained,  the 
kinetic  energy  becomes  a  quadratic  function  with  constant  coefficients.  Hence  the 
Lagrangian  function  (cf .  §  160) 

L=T-V=  T{q, ,  r/„ ,  .  . . ,  q„)  -  V{q, .  q, ,  •  •  • ,  r/„), 
when  substituted  in  the  formulas  for  the  motion  of  the  system,  gives 
d  cL       cL  _  d  cL       cL  _  d  cL       ZL  _ 

dt  cq^       cqy  dt  c('i.2       cq.2  dt  cq„       cq,, 

a  set  of  equations  of  the  second  order  with  constant  coefficients.  The  equations 
moreover  involve  the  operator  D  only  through  its  square,  and  the  roots  of  the  equa- 
tion in  D  must  be  either  real  or  pure  imaginary.  The  pure  imaginary  roots  intro- 
duce trigonometric  functions  in  the  solution  and  represent  vibrations.  If  there  were 
real  roots,  which  would  have  to  occur  in  pairs,  the  positive  root  would  represent 
a  term  of  exponential  form  which  would  increase  indefinitely  with  the  time,  —  a 
result  which  is  at  variance  both  with  the  assumption  of  stable  equilibrium  and 
with  the  fact  that  the  energy  of  the  system  is  constant. 

"When  there  is  friction  in  the  system,  the  forces  of  friction  are  supposed  to  vary 
with  the  velocities  for  small  vibrations.  In  this  case  there  exists  a  dissipative  func- 
tion F{qy^,  q„,  •  •  • .  q„)  which  is  quadratic  in  the  velocities  and  may  be  assumed  to 
have  constant  coefficients.    The  equations  of  motion  of  the  system  then  become 

d  cL       cL       cF  _  d  cL       cL       cF  __ 

dt  cq^        cqy        cq^  dt  cq,,        cq,,       cq,, 

which  are  still  linear  with  constant  coefficients  but  involve  first  powers  of  the 
operator  D.  It  is  physically  obviovis  that  the  roots  of  the  etjuation  in  JJ  must  be 
negative  if  real,  and  must  have  their  real  parts  negative  if  the  roots  are  complex  ; 
for  otherwise  the  energy  of  the  motion  would  increase  indefinitely  with  the  time, 
wliereas  it  is  known  to  be  steadily  dissipating  its  initial  energ\-.  It  may  be  added 
that  if,  in  addition  to  tlie  internal  forces  arising  from  the  potential  V  and  the 


22^  DIFFERENTIAL  EQUATIONS 

frictional  forces  arising  from  the  dissipative  function  F,  tliere  are  other  forces 
impressed  on  the  system,  these  forces  wovild  remain  to  be  inserted  upon  tlie  riglit- 
hand  side  of  the  equations  of  motion  just  given. 

Tlie  fact  tliat  tlie  e(iuations  for  small  vibrations  lead  to  equations  with  constant 
coefficients  by  neglecting  the  higher  powers  of  the  variables  gives  the  important 
physical  theorem  of  the  superposition  of  small  vibrations.  The  theorem  is  :  If  with 
a  certain  set  of  initial  conditions,  a  sy.stem  executes  a  certain  motion  ;  and  if  with 
a  different  set  of  initial  conditions  taken  at  the  same  initial  time,  the  system 
executes  a  second  motion  ;  then  the  system  may  execute  the  motion  which  consists 
of  merely  adding  or  superposing  these  motions  at  each  instant  of  time  ;  and  in 
particular  tin's  combined  motion  will  be  that  which  the  system  would  execute  under 
initial  conditions  which  are  found  l)y  simply  adding  the  corresponding  values  in 
the  two  sets  of  initial  conditions.  This  theorem  is  of  course  a  mere  corollary  of  the 
linearity  of  the  equations. 

EXERCISES 

1.  Integrate  tlie  following  systems  of  equations  : 

{(x)  I)x  —  Dij  +  X  =  cos  i,  D'^x  —  By  4.  3  x  —  ?/  =  e^ «, 

(/3)  3  l)x  +  3  X  +  2  //  =  e',  4  x  -  3  Ihj  +  3  //  =  3  <, 

(7)  D2x  —  3x  —  4?/  =  0,  IPlJ  +  X  +  ?/  =:  0, 

(8)  = =  clt,  (e)  —  (It  = = , 

y  —  7x      2 X  +  5 //  3 X  +  4 //       2 x  +  5 y  • 

(f)  //>x  +  2  (x  -  ?/)  =  1,  tJ)y  +  X  +  •')//  =  t, 

{t})  I)x  =  ny  —  ?H2,  By  =  Iz  —  ?)X,  Bz  =  mx  —  ly, 

(0)  B-^x  -  3  X  -  4  ?/  +  3  =  0,  -D2//  +  x  -  8  7/  +  5  r=  0, 

( 1 )  B^x  -  4  Ifiy  +  4  B'^x  -  x  =  0,  B^y  -  4  B"x  +  4  Bhj  -  //  =  0. 

2.  A  particle  vibrates  without  friction  upon  the  inner  surface  of  an  ellipsoid. 
Discuss  the  motion.    Take  the  ellipsoid  as 

i.  +  'L-i-i:: ^  =  1;     then     x  =  C'sin    — ^' i  +  c;,    ,     ,/  =  irsin(--^i  +  iv 

3.  Same  as  Ex.  2  when  friction  varies  with  the  velocity. 

4.  Two  heavy  particles  of  eijual  mass  are  attached  to  a  light  string,  one  at  the 
middle,  one  at  one  end,  and  are  suspended  by  attacliing  the  other  end  of  the  string 
to  a  fixed  point.  If  the  particles  are  slightly  displaced  and  tlie  oscillations  take 
place  without  friction  in  a  vertical  jilane  containing  the  fixed  point,  discuss  the 
motion. 

5.  If  there  be  given  two  electric  circuits  witliout  capacity,  the  ecjuations  are 

-r    dU        ,rdi„       T^  .         T-,  T    '''■.        ,,di,        ,.  . 

^  dt  dt  ^^         ^ '  -  dt  dt  '  ' 

where  ij ,  i„  are  the  currents  in  the  circuits,  ij ,  L.,  arc  tlic  coefficients  of  solf- 
induction,  /i'^,  /.'._,  are  the  resistances,  and  M  is  the  coefficient  of  nuitual  induction, 
(cr)  Integrate  the  ecjuations  wlien  tlie  inqjressed  electromotivt'  forces  2?j,  E„  are 
zero  in  both  circuits.  (/3)  Also  when  E.^  =  0  but  £",  =  sin  pi  is  a  x>eriodic  force. 
(7)  Discuss  the  cases  of  loose  coupling,  that  is,  where  M'^/L^L.,  is  small ;  and  the 
case  of  close  coupling,  that  is,  wlierci  M'^/L^L.,  is  nearly  unity.  What  values  forj) 
are  especially  notewortliy  wlien  the  damping  is  small  ? 


COMMOXEJl  ORDIXARY  EQUATIONS  2:^7 

6.  If  the  two  circuits  of  Ex.  5  liave  capacities  C^,  C.^  and  if  q^,  q.^  are  the 
cliarges  on  the  condensers  so  that  i^  =  dq^/dt,  t.^  =  dq.^/dt  are  the  currents,  the 
equations  are 

d^       ^^d^  dq,        q^^  d%,       ^^d^  Clq,        9^  ^  ^ 

^  dt^  dfi  ^  dt       C^         ^  ^  dt^  dt^  ~  dt       C.^         " 

Integrate  when  the  resistances  are  negligible  and  Ei=  E„=  0.  If  T^  =  27r  VC^i^ 
and  T.^  =  2irVC2L.^  are  the  periods  of  the  individual  sejjarate  circuits  and 
e  =  2^J/VC\C'J  and  if  T^  =  T.,,  show  that  VT"^  +  62  and  Vt^  -  B'-^  are  the 
indejjendent  periods  in  the  coux^led  circuits. 

7.  A  uniform  beam  of  weight  0  lb.  and  length  2  ft.  is  placed  orthogonally 
across  a  rough  horizontal  cylinder  1  ft.  in  diameter.  To  each  end  of  the  beam  is 
suspended  a  weight  of  1  lb.  upon  a  string  1  ft.  long.  Solve  the  motion  produced 
by  giving  one  of  the  weights  a  slight  horizontal  velocity.  Note  that  in  finding  the 
kinetic  energy  of  the  beam,  the  beam  may  be  considered  as  rotating  about  its 
middle  point  (§  39). 


CITAPTEIf  IX 
ADDITIONAL  TYPES  OF  ORDINARY  EQUATIONS 

100.  Equations  of  the  first  order  and  higher  degree.  The  degree  of 
a  dittV'reiitial  equation  is  defined  as  the  degree  of  the  derivative  of 
higliest  oi'der  wliieli  enters  in  the  equation.  In  the  ease  of  the  equation 
^(.r,  I/,  '/)  =  0  of  tlie  iirst  order,  the  degree  will  he  the  degree  of  the 
equation  in  y'.  From  the  idea  of  the  lineal  element  (§  85)  it  appears 
that  if  the  degree  of  ^  in  y'  is  n,  there  will  be  n  lineal  elements  through 
each  point  (./•,  y).  Hence  it  is  seen  that  there  are  n  curves,  which  are 
compounded  of  these  elements,  ])assing  through  each  point.  It  may  be 
pointed  out  that  equations  such  as  //'  =  .rVl  +  //-,  wliich  are  apparently 
of  the  first  degree  in  //',  are  really  of  higher  degree  if  the  multiple  value 
of  the  functions,  such  as  Vl  +  //-,  which  enter  in  the  equation,  is  taken 
into  consideration;  the  equation  aljove  is  replaceable  by  y'-  —  x-  -\-  .ry-, 
which  is  of  the  second  degree  and  without  any  niidtiple  valued  function.* 

First  suppose  that  tlie  d'lffi'rcnfhd  eiitiat'ion 

*  (■'•,  !h  !/')  =  [//'  -  '/'/■'■,  //)]  X  [//'  -  lAi.-'S  .V)]  •  •  •  =  0  (1) 

7nay  be  soJred  fnr  y'.    It  then  bt^comes  cipiivalent  to  the  set 

//'  -  ^pp^,  y)  =  0,  //'  -  ./../,/•,  //)  =  0,  •  •  •  (1') 

of  equations  each  of  the  first  ord(^r,  and  each  of  these  may  be  treated 
by  the  methods  of  Chap.  VIII.    Thus  a  set  of  integrals  t 

J-\(,';  ]h  n  =  0,  Fj.r,  y,  n  =  0,  .  .  •  (2) 

may  be  obtained,  and  the  })roduct  of  these  separate  integrals 

F(.r,  //,   r)  =  F^i.r,  y,  C)  ■  Fj,',  y,  C)  •  •  •  =  0  (2') 

is  the  complete  solution  of  the  original  e(]uati()n.  (Tcometrically  speak- 
i]ig,  each  integral  /•',('';  //•  '")=  0  represents  a  family  of  curves  and  the 
]>roduct  represents  all  the  i'aniilics  simultaneously. 

*  It  is  tliiTi'forc  apparent  tliat  tin-  idea  of  df^rce  as  aiii>lir(l  in  praftic*-  is  Sdincwliat 
inilefinitf. 

t  Tlic  same  constant  '''  or  any  desircil  function  of  ''  may  be  used  in  the  ditTerent 
solutions  because  ('  is  an  arbitrary  constant  and  no  specialization  is  introiluced  by  its 
repeated  use  in  this  way. 


ADDITIONAL  OKDIXARY  TYPES  229 

As  an  example  consider  y'^  +  2  y'y  cot  x  =  y-.    Solve. 

y'^  +  2  y'y  cot  x  +  y-  cot^  x  =  y^l  +  cot^  x)  =  y^  csc^  x, 
and  (y'  +  y  cot  x  —  ?/  esc  x)  {>/  +  y  cot  x  +  ?/  esc  x)  =  0. 

These  equations  both  come  under  the  type  of  variables  sei:)arable.    Integrate 

dy      1  —  cos  X  d  cos  x 


y  sinx  1  +  cosx 


y{l  +  cosx)  =  C, 


dw  1  4-  cosx  ,  (Zcosx  ,,  ,       „ 

and  —  = dx  = ,        2/  (1  —  cosx)  =  6. 

2/  sinx  1  —  cosx 

Hence  [y  (1  +  cos x)  +  C]  [ij  (1  -  cos  x)  +  t']  =  0 

is  the  solution.    It  may  be  put  in  a  different  form  hy  nuiltiplying  out.    Then 

i/-sin2x  +  2  CV  +  C"^  =  0. 

If  the  equation  cannot  be  solved  for  y'  or  if  the  equations  resulting 
from  the  solution  cannot  be  integrated,  this  first  method  fails.  In  that 
case  it  may  he  j^ossUde  to  solce  for  y  or  for  x  and  treat  the  equation  by 
dilferentiation.    'hat  y'  =p.    Then  if 

The  equation  thus  found  by  differentiation  is  a  differential  equation  of 
the  first  order  in  dp/dx  and  it  may  be  solved  l)y  the  methods  of  Chap. 
VIII  to  find  F(^2^,  x,  C)  =  0.    The  two  equations 

y=f{x,2>)     and     Fip,x,r)=={)  (3') 

may  be  regarded  as  defining  x  and  y  parametrically  in  terms  oi  p,  or  2^ 
may  be  eliminated  between  them  to  determine  tlie  solution  in  tlie  form 
Q,  (x,  y,  C)  =  0  if  this  is  more  convenient.  If  the  given  differential  equa- 
tion had  been  solved  for  x,  then 

'■''=/ (U,  J')     and     -y-=  — =  — 4-— -— •  (4) 

The  resulting  equation  on  the  right  is  an  equation  of  the  first  order  in 

d2j/<hj  and  may  be  treated  in  the  same  way. 

As  an  example  take  xp-  —  2  yp  +  WJ"  =  0  and  sf)lve  for  //.    Then 

ax  ^dy      ^  ,      dp      ax  dp   ,  a 

2y  =  xp+^,         2-f  =  2p=p  +  x^---^  +  -, 
p  dx  dx      p^  dx      p 

or  —\p  —  —  \  —  -\-{- P  I  =  0.     <^'i"    -i-'h'  —  P'^l-^  =  0- 


p\_         pjdx      \p        I 


The  solution  of  this  equation  is  x  =  Cp.    The  solution  of  the  given  equation  is 

2  y  =  xp  H ,         X  —  Cp 

P 

when  expressed  parametrically  in  terms  of  p.    It  p  be  eliminated,  then 

2y  =  —  +  aC  parabolas. 

G 


230  DIFFEllEXTIAL   EQUATIONS 

As  another  example  take  'p-y  +  2  j>x  =  y  and  solve  for  x.    Then 

\V        I  dy      p      p       ■  \      p^         /  dy 

or  -  +  p  +  ?/  (—  +  1 )  -—  =  0,     or     ?/Jjj  +  jKhj  =  0. 

p  \p-        /  dy 

The  solution  of  this  is  j)y  =  C  and  the  solution  of  tlie  given  equation  is 
2  X  =  ?/  / p  j  ,       py  =  C,     or    ?/2  =  2  C'x  +  C'^. 

Two  special  types  of  equation  may  be  mentioned  in  addition,  altliough 
tlunr  method  of  s(dution  is  a  mere  corollary  of  the  methods  already 
given  in  general.  They  are  the  equation  ]uniiognneotis  in  (./',  y)  and 
Clalraufs  equation.  The  general  form  of  the  homogeneous  equation  is 
^{I'i  u/-'')^  ^-    This  equation  may  he  solved  as 

P  =  ^(^  or  as     l=f(j^         V  =  ^''f{v)\  (5) 

and  in  the  first  case  is  treated  l)y  the  methods  of  ("luip.  VIII,  and  in 
the  second  by  tlu'.  nu'thods  of  this  article.  Which  method  is  cliosen 
rests  with  the  solver.    The  Clairaut  type  of  ecpiation  is 

y=l'^'rf{p)  (6) 

and  comes  directly  under  the  methods  of  this  article.  It  is  esj^ecially 
noteworthy,  however,  that  on  differentiating  with  res})ect  to  x  the  result- 
ing equation  is  i  , 

[«+/(.')]  ;£=o  0.  ;^;=o.  (6') 

Hence  the  solution  for  ^^  is  j)  =  C,  and  thus  ?/  =  Cx  +,/'('^')  is  the  solu- 
tion for  the  Clairaut  equation  and  represents  a  family  of  straight  lines. 
The  rule  is  merely  to  substitute  C  in  })lace  of  y/.  This  tv2)e  occurs  very 
frequently  in  geometric  ap})lications  either  directly  or  in  a  disguised 
form  requiring  a  preliminary  change  of  variable. 

101.  To  this  point  the  only  solution  of  tin;  differential  equation 
^(x,  ]/,  yy)  =  0  which  has  been  considered  is  the  (/rwrnl  sa/i/fion 
F(x,y,  (''j=Q  containing  an  arbitrary  constant.  If  a  sjiecial  value, 
say  2,  is  given  to  C,  the  solution  F(x,  //,  2)  =  0  is  called  a.  jHirtifiihir 
solution.  It  may  ha])})en  that  the  arbitrary  constant  ('  eiitei's  into  the 
expression  /''(•'",  //,  '")  =  0  in  such  a  Avay  that  when  ('  becomes  jiositively 
infinite  (or  negatively  infinite)  the  curve  F{x,  //,  '")=  0  ap])roaches  a 
definite  limiting  ])Osition  which  is  a.  solution  of  the  diifei'eiitial  ('(puition  ; 
such  solutions  are  called  intinifr.  sohifians.  In  addition  to  tlu'se  types 
of  solution  which  naturally  grou})  themselves  in  connection  witli  the 
general  solution,  there  is  often  a  solution  of  a  differ(Mit  kind  which  is 


ADDITIONAL   OiiDIXAHY   TYPES  231 

known  as  the  si/ujular  t>olutlon.  There  are  several  different  delinitions 
for  the  singidar  solution.  That  which  will  be  adopted  here  is  :  A  singu- 
lar solution  is  the  encelope  of  the  family  of  curves  defined  hij  the 
(jeneral  solution. 

The  consideration  of  the  lineal  elements  (§  85)  will  show  how  it  is 
that  the  envelope  (§  65)  of  the  family  of  particular  solutions  which 
constitute  the  general  solution  is  itself  a  solution  of  the  equation.  For 
consider  the  figure,  which  represents  the  particular  solutions  broken  up 
into  their  lineal  elements.  Kote  that  the  envelope  is  made  up  <jf  those 
lineal  elements,  one  taken  from  each  })articular  so- 
lution, which  are  at  the  points  of  contact  of  the  envelope 
envelope  with  the  curves  of  the  famih'.  It  is  seen  ^f^  xT^ 
that  the  envelope  is  a  curve  all  of  whose  lineal 
elements  satisfy  the  equation  ^  (,/■,  y,  p)  =  0  for  the 
reason  that  they  lie  upon  solutions  of  the  e(|uation.  Xow  any  curve 
whose  lineal  elements  satisfy  the  equation  is  by  definition  a  solution 
of  the  equation ;  and  so  the  envelope  must  be  a  solution.  It  might 
conceivably  happen  that  the  family  FQr,  >/,  C)=  0  was  so  constituted 
as  to  envelope  one  of  its  own  curves.  In  that  case  that  curve  would 
be  both  a  particular  and  a  singular  solution. 

If  the  general  solution  F(x,  y,  (')  =  0  of  a  given  differential  equation 
is  known,  the  singular  solution  niay  be  found  according  to  the  rule  for 
finding  envelopes  (§  Go)  by  eliminating  C  from 

F(,r.  y,  C)  =  0      and      r^  F(,r,  y,  C)  =  0.  (7) 

It  should  be  borne  in  mind  that  in  the  climinant  of  these  two  equations 
there  may  occur  some  factors  which  do  not  represent  envelopes  and 
which  must  l)e  discarded  from  the  singular  solution.  If  only  the  singu- 
lar solution  is  desired  and  the  general  solution  is  not  known,  this 
method  is  inconvenient.  In  the  case  of  Clairaut's  equation,  however, 
where  the  solution  is  known,  it  gives  the  result  immediately  as  that 
obtained  b}'  eliminating  C  from  the  two  equations 

y  ^  Cx  +f(r)     and     0  =  ,■  +f{r).  (8) 

It  may  l)e  noted  that  as  p  =  C,  the  second  of  the  equations  is  merely 
the  factor  ,/•  +. /"'(/')  =  0  discarded  from  (G').  The  singular  solution  may 
therefore  be  found  by  eliminating  p  Ijetween  the  given  Clairaut  e(|ua- 
tion  and  the  discarded  factor  ./■  +/''(y/)=  0. 

A  reexamination  of  the  figure  will  suggest  a  means  of  finding  the 
singular  solution  without  integrating  the  given  equation.  For  it  is  seen 
that  v/lu'ii  two  neighboring  curves  of  the  familv  intersect  in  a  ]joint  P 


232  DIFFERENTIAL  EQUATIONS 

near  the  envelope,  then  through  this  point  there  are  two  lineal  elements 
which  satisfy  the  differential  equation.  These  two  lineal  elements  have 
nearly  the  same  direction,  and  indeed  the  nearer  the  two  neighboring 
cm'ves  are  to  each  other  the  nearer  will  their  intersection  lie  to  the 
envelope  and  the  nearer  will  the  two  lineal  elements  approach  coinci- 
dence with  each  other  and  with  the  element  upon  the  envelope  at  the 
point  of  contact.  Hence  for  all  points  (x,  //)  on  the  envelope  the  equa- 
tion ^{x,  y,  jj)—  0  of  the  lineal  elements  must  have  douhJe  roots  for  p. 
Now  if  an  equation  has  double  roots,  the  derivative  of  the  equation 
must  have  a  root.    Hence  the  requirement  that  the  two  equations 

"A  (•''.  y,  P)  =  0     and     ^  xp  {x,  ;/,  p)  =  0  (9) 

Cp 

have  a  common  solution  for  p  Avill  insure  tliat  the  first  has  a  double 
root  for  2' ',  and  the  })oints  (,'•,  >/)  which  satisfy  these  equations  simul- 
taneously must  surely  include  all  the  points  of  the  envelope.  The  rule 
for  finding  the  singular  solution  is  therefore:  Ell m Inatc  p  from,  the 
fjlren  (I!fferf'7itial  equation  and  Its  (Jerlrcitice  with  respert  to  p,  that  is, 
from  (9).    The  result  should  be  tested. 

If  tlie  equation  xp-  —  2  yp  +  ax  =  0  treated  above  be  tried  for  a  singular  solution, 
the  elimination  of  p  is  required  between  the  two  equations 

xp-  —  2  i/p  +  ux  =  0     and     xp  —  y  =  0. 

The  result  is  y-  =  ax'-,  which  gives  a  pair  of  lines  through  the  origin.  The  substi- 
tution of  y  =  ±  \/ax  and  p  =  ±  Va  in  the  given  equation  shows  at  once  that 
y-  =  ox'-^  satisfies  the  equation.  Thus  y-  =  ax-  is  a  singular  solution.  The  same 
result  is  found  by  finding  the  envelope  of  the  general  solution  given  above.  It  is 
clear  that  in  this  case  the  singular  solution  is  not  a  particular  solution,  as  the  par- 
ticular .solutions  are  parabolas. 

If  the  elimination  had  been  carried  on  by  Sylvester's  method,  then 

0  X     —  y\ 

X     —  2  y  a\=  —  X  (y-  —  ax-)  =  0  ; 

X     -     y  o| 

and  the  eliminant  is  the  product  of  two  factors  .r  =  0  and  //-  —  ax-  =  0.  of  which 
the  second  is  that  just  found  and  the  first  is  the  (/-axis.  As  the  slope  of  the  y-ax'ia 
is  infinite,  the  substitution  in  the  e(]uation  is  hardly  legitimate,  ami  the  eiiuation 
can  hardly  be  said  to  be  satisfied.  The  occurrence  of  these  extraneous  factors  in 
the  eliminant  is  the  real  reason  for  the  necessity  of  testing  the  result  tn  see  if  it 
actually  represents  a  singular  solution.  These  extraneous  factors  may  reju'esent 
a  great  variety  of  conditions.  Thus  in  the  case  oi  the  equation  p/-  +  2  yp  cot  x  =  y- 
previously  treated,  the  elimination  gives  y-  csc'-.f  =  0.  and  as  esc  x  cannot  vanish. 
the  result  reduces  to  y-  =  0.  or  the  j"-axis.  As  the  slope  along  the  j"-axis  is  0  ami  y 
is  0,  the  eijuation  is  clearly  satisfied.  Yet  the  line  y  =  0  is  not  the  envelope  of  the 
general  solution  ;  for  the  curves  of  the  family  touch  the  line  oidy  at  the  points  rnr. 
It  is  a  particvdar  solution  and  corresponds  to  C  —  0.    There  is  no  singular  solution. 


ADDITIONAL  OKDINAEY  TYPES  233 

Many  authors  use  a  great  deal  of  time  and  space  discussing  just  what  may  and 
what  may  not  occur  among  the  extraneous  h)ci  and  how  many  times  it  may  occur. 
The  result  is  a  considerable  number  of  statements  which  in  their  details  are  either 
grossly  incomplete  or  glaringly  false  or  both  (cf.  §§  65-67).  The  rules  here  given 
for  finding  singular  solutions  should  not  be  regarded  in  any  other  light  than  as 
leading  to  some  expressions  which  are  to  be  examined,  the  best  way  one  can,  to 
find  out  whether  or  not  they  are  singular  solutions.  One  curve  which  may  appear  in 
the  elimination  of  p  and  which  deserves  a  note  is  the  tac-locus  or  hicus  of  points  of 
tangency  of  the  particular  solutions  with  each  other.  Thus  in  the  system  of  circles 
{x  —  C)'^  +y"^  =  r-  there  may  be  found  two  which  are  tangent  to  each  other  at  any 
assigned  point  of  the  x-axis.  This  tangency  represents  two  coincident  lineal 
elements  and  hence  may  be  expected  to  occur  in  the  elimination  of  p  between  the 
differential  equation  of  the  family  and  its  derivative  with  respect  to  p  ;  but  not  in 
the  eliminaut  from  (7). 

EXERCISES 

1.  Integrate  the  following  e(iuations  by  solving  for  p  =  i/-. 

(a)  p^  _  6p  +  5  =  0,  (/3)  p^  -  {2x  +  i,'-^)p-  +  {x^  -  if-  +  2  jy-^)p-  (x^  _ ^2)^2^0, 

(7)  a;p2  _  2  //p  -  X  =  0,        (5)  p'  (X  +  2 ;/)  +  3p-^  (x  +  ,j)  +  p  {y  +  2  x)  =  0, 

(e)  y^  +  p^  =  1,  (f)  p-^  -  «x3  =  0,  (7,)  p  =  {«  _  X)  Vl  +  p-^. 

2.  Integrate  the  following  equations  by  S(jlving  for  (/  or  x  : 

(a)  4xp2  j^  2xp  -  v/  =  0,  (/3)  y  =  -  xp  +  x*p-,  (7)  p  +  2 xy  -  x^  -  2/2  =  0, 

(5)  2px  —  ?/  +  logp  =  0,  (f)  -r  —  yp  =  ap',  {^)  y  =  x  +  a  tan-ip, 
{t))  x  =  y  +  a  logp,  (6)  x  +  py  {2p"  +  3)  =  0,  ( t)  a-yp-  -  2  xp  +  y  =  0, 
(k)  p^  —  ixyp  +  8  (/-  =  0,  (X)  X  =  p  +  logp,  (fj.)  p-(x~  +  2  ax)  =  «-. 

3.  Integrate  these  e(juations  [substitutions  suggested  in  (t)  and  (k)]  : 

(a)  xy-  {p-  +  2)  =2p^3  ^  ^s^  (^^  („j.  ^  ^^^2  =  (i  +  p"-)  (f  ^  ,,^.2)^ 

(7)  y"  +  -ryp  —  a;V'  =  o,  (5)  .'/  =  yp-  +  2px, 

( f )  y  =  px  +  sin-ip,  (f )  //  =  p (x  —  h)  +  rt/p, 

(7?)  y  =px-\-  p  (1  -  p-).  {6)   y"  -  -Ipxy  —  1  -  p-  (1  -x'-), 

(t)  ie-'Jp-  +  2xp  —  1  =  0,    z  =  e-!',  (k)  y  =  2 px  +  y-p".    y-  —  z, 

(X)  Ae'^yp-  +  2e--'-p  —  e-'^'  =  0,  (m)  x-  (y  —  px)  =  //p-. 

4.  Treat  these  e<juations  by  the  p  metliod  (U)  to  lind  tlie  singular  solutions. 
Also  solve  and  treat  by  the  C  method  (7).  Sketch  the  family  of  solutions  and 
examine  the  significance  of  the  extraneous  factors  as  well  as  that  (tf  the  factor 
which  gives  the  singular  solution  : 

(or)  p2(/  +  p  (x  —  (/)  —  X  =  0,         ((3)  fry-  cos-  a  —  2px//  sin-  or  +  //-  —  x-  sin"  a  =  0, 
(7)  4 xp2  =  (3  X  —  a)-,  ( 5)  yp-x  (x  —  a)  (x  -  h)  =  [3 x-  —  2  x  (a  +  Ji)  +  «'']-, 

( 6 )  p2  +  xp  -  ^  =  0,  ( f )  8  (M 1  +  pY  =  27  (X  +  y)  (1  -  p)^, 
(t?)  x3p--  -F  ^-yp  +  «='  =  0,  (^)  z/  (:5  -  4  y)2p--  =  4(1-//). 

5.  Examine  sundry  of  the  ecjuations  of  Exs.  1.  2.  3,  for  singular  solutions. 

6.  Show  that  the  solution  of  y  =  X(p(p)  -f-/(p)  is  given  paranietricaliy  by  the 
given  equation  and  the  solution  of  tlie  linear  equation: 

—  -I-  X  — ^-'^-^  -  -  •   '  V' Solve     (a)   II  =  vixp  +  ?i  (1  +  !>')-, 

dp         -i&(p)-p P^-4>{p) 

(^)   y  =  X  (p  +  «  Vr+  P-),  (7)  -t  =  yp+  tqi-,  (5)   //  =  (1  +  p)  x  +  P^. 


234  DIFFEKEXTIAL  EQUATIONS 

7.  As  any  straight  line  is  y  =  mx  +  h.  any  family  of  lines  may  be  represented  as 
y  =  nix  +f{m)  or  by  the  Clairant  eciuatiun  y  =  px  +/{]>).  Show  that  the  orthog- 
onal ti'ajectories  of  any  family  of  lines  leads  to  an  ecjnation  of  the  type  of  I-^x.  (i. 
The  same  is  true  of  the  trajectories  at  anj-  constant  angle.  Express  the  eciuations 
of  the  following  systems  of  lines  in  the  Clairaut  form,  write  the  equations  of  the 
orthogonal  trajectories,  and  integrate  : 

{a)  tangents  to  a-^  +  ^/^  =  1,  (|3)  tangents  to  y^  =  2rtj, 

(7)  tangents  to  y^  =  x^,  (5)  normals  to  y-  =  2ux, 

(e)  normals  to  y-  =  x^,  (f)  normals  to  h-x-  +  ii-y-  =  a-h-. 

8.  The  ctolute  of  a  given  curve  is  the  locus  of  the  center  of  curvature  of  the 
curve,  or,  what  amounts  to  the  same  thing,  it  is  the  envelope  of  the  normals  of  the 
given  curve.  If  the  Clairaut  ecjuation  of  tlie  normals  is  known,  the  evolute  may  be 
obtained  as  its  singular  solution.    Thus  tind  the  evolutes  of 

{a)  y'^  =  iax,  (/3)  2xy  =  u"^,  (7)  X3  +  ys  =  as, 

a-      0-  2(1  —  X 

9.  The  involutes  of  a  given  curve  are  the  curves  which  cut  the  tangents  of  the 
given  curve  orthogonally,  or,  what  amounts  to  tlie  same  thing,  tlie}'  are  the  curves 
which  have  the  given  curve  as  the  locus  of  their  centers  of  curvature.  Find  the 
involutes  of 

{a)  XT  -{■  y-  =  a2,  (/3)  y-  =  2  mx,  (7)  y  =  a  cosh  (//((). 

10.  As  any  curve  is  the  envelope  of  its  tangents,  it  follows  that  when  the  curve 
is  described  by  a  property  of  its  tangents  the  curve  may  be  reganled  as  the  singu- 
lar solution  of  the  Clairaut  Ciptation  of  its  tangent  lines.  Dett-rnune  thus  what 
curves  have  these  properties  : 

(a)  length  of  the  tangent  intercepted  Ijetween  the  axes  is  I, 

{(3)  sum  of  the  intercepts  of  the  tangent  on  the  axes  is  c, 

(7)  area  between  the  tangent  and  axes  is  the  constant  A'-, 

(5)  product  of  i)erpendicuiars  from  two  fixed  points  to  tangent  is  k-, 

(e)  product  of  ordinates  from  two  points  of  x-axis  to  tangent  is  A'-. 

(IF  I 

11.  From  the  relation  —  =  fx  vJ/-  -f-  .V-  of  Proposition  3.  p.  212.  show  that  as 

dn 

the  curve  F  =  C  is  moving  tangentially  to  itself  along  its  envelope,  the  singular 
solution  of  ^Ulx  +  -V'7//  =  0  may  l)e  expected  to  be  found  in  the  eijuation  1/^t  =  0  : 
also  tlie  infinite  solutions.    Discuss  the  e(juation  \/p.  —  0  in  tlu-  following  cases: 


((()    X  1  —  y-dx  =  X  1  —  x-dy.         {,i}  xdx  +  ydy  =  \  x-  +  y-  —  n-dy. 

102.  Equations  of  higher  order.    In  i\w  troatnu-nt  of  siieciul  }»rol> 
leins  (§  8l'j  it  was  seen  that  tlu*  siil:)stitiitioiis 

rendered  the  differential  erjuations  integral )le  by  reducing  them  to  in- 
tegrahle  ecitiations  of  the  first  order.  These  substitutions  or  othei'S  like 
them   ai-e  tiseful   in  treating  certain   eases  of  the  diiferential  eijuation 


ADDITIONAL  ORDINARY   TYPES  235 

<if(x,  y,  y',  y",  ■■■,  y^"))  =  0  of  the  nth  onk-r,  namely,  when  one  of  tlie 
variables  and  perliaps  some  of  the  derivatives  of  lowest  order  do  not 
occur  in  the  equation. 

In-se  *(^,,_-^,   _;(,...,  _=^)  =  o,  (11) 

y  and  the  first  i  —  1  derivatives  being  absent,  substitute 

g  =  ,     so...     .(,„|,....p)  =  0.  (11, 

The  original  equation  is  therefore  replaced  by  one  of  lower  order.  If 
the  integral  of  this  be  F(:r,  ^)  =  0,  which  will  of  course  contain  n  —  I 
arbitrary  constants,  the  solution  for  q  gives 


y  =/(,.)      and     y=j...jf{.r){dxy. 


(12) 


The  solution  has  therefore  been  accomplished.    If  it  were  more  con- 
venient to  solve  -P(.'',  7)  =  0  for  .?•  =  4>('/),  the  integration  would  be 


1/ 


=j. .  J  .y  (d.ry  =  J.  .  .J.J  [<^'(  y)  d^y  ;  ^  (12') 

and  this  equation  with  ,r  =  <^(V/)  -would  give  a  parametric  expression 
for  the  integral  of  the  differential  equation. 

,(,.g,g,..,£.).0,  (13) 


X  being  absent,  substitute  ])  and  regard  p  as  a  function  of  y.    Then 
dy  d'y  dp  d^y  d  I     d j 


dx    ^^^'       7i?  =  ^';n/       'd?-J';n,[''dy^ 

and  ^,^y,^>,  -,...,  ^  =  0. 

In  this  way  the  order  of  the  equation  is  lowered  by  unity.  If  this  equa- 
tion can  be  integrated  as  F(y,  ji)  =  0,  the  last  stfq)  in  the  solution  may 
be  obtained  either  directly  or  parametiically  as 

It  is  no  particular  simplification  in  this  case  to  have  some  of  the  lower 
derivatives  of  y  absent  from  ^  =  0,  l)ecause  in  general  the  lower  deriva- 
tives of  ])  will  none  the  less  be  introduced  by  the  substitution  that 
is  made. 


236  DIFFERENTIAL   EQUATIONS 

As  ail  example  coiikuUt  ( x  — '■ ^  )  =  (  —     +  1» 

\    (/x>      dx-/        \(lx'/ 

which  is  ( X  -i  -  f/    =    ~     +  1     it     (/  =  —^  • 

\   dx        /       \dx/  dx- 

Tlieii  g  =  X  —  ±  A I  (-^\  +  1     and    q  =  C.x  i  Vc',-^  + 1 ; 

dx         >  \dx/ 

for  the  equation  is  a  Clairaut  type.    Hence,  finally, 

y  -^fj\_GxX,  ±  Vc'i-  +  l](dx)2  =  1  C^x^  ±  ia;2  Vc'f  +  f  +  C^x  +  Cg. 

As  another  example  consider  y"  —  ?/'2  _  ^2  log?/.    This  becomes 

p-^—p-=y^  log  2/     or     — \-^  -  2p-  =  2  ?/-  log  ?/. 
dy  dy 

The  eqiiation  is  linear  in  jr  and  lias  the  integrating  factor  t'--''. 

-  p^c-  ■-'  2/  =  j  ^2e-  - ^  log  yd?/,         -—  p  =    c'-i  !i  fy-c-  '^  v  log  ydy      , 

and  /  ^ j-  =  V2x. 

I      e2  J/  r?/2e-  2 ."  log  ydy 

The  integration  is  therefore  reduced  to  quadratures  and  becomes  a  problem  in 
ordinary  integration. 

If  an  equation  is  iKiivogcncnua  icltJi  resj'x'ct  to  y  and  its  derivatives, 
that  is,  if  the  e(|uation  is  niultipliinl  by  a  power  of  h  when  y  is  replaced 
by  />•//,  the  order  of  the  equation  may  l)e  lowered  by  the  substitution 
y  =  ('~'  and  Viy  taking-  z'  as  the  new  varialile.  If  the  equation  is  hnmn- 
(jnii'oiis  irith.  respect  to  x  and,  dx,  that  is,  if  tlie  equation  is  multiplied 
by  a  ])0wer  of  /.■  wlien  ,/•  is  replaced  by  /.■;*•,  the  oi'der  of  the  equation 
may  lie  reduced  by  tlu^  substitution  ,/•  =  «'.  The  work  may  be  simplified 
(Ex.  9,  p.  152)  by  the  use  of 

/>;//  =  <'r-ri),{D,  -  1)  . . .  {I),  -n  +  V)y.  (15) 

If  the  equation  is  liomof/cneoiis  iritli  respeet  to  x  and  y  and,  the  dif- 
ferentials dx,  dy,  d'^i/,  ■  ■  ■ ,  tlu!  ordiT  may  be  lowered  by  the  substitution 
X  =  e',  y  =:  e'z,  wliei'e  it  may  be  recalled  that 

irjy  =  r-"' />//>,  -  1)  .  • .  (I\  -  7.  +  1)// 

=  ,--("-iV(/;^  +  1)  />,  •  •  •  {'K  -n  +  2)::.  ^  '  ^ 

Finall}',  if  tlie  e(iuation  is  /nu/iof/eneoi/s  iritli  respect  to  x  considered  of 
divieiisions  1,  and  y  considered,  of  dimensions  ni,  that  is,  if  tlu;  equation 
is  multiplied  by  a  ])<)\vcr  of  /,•  wlicn  /.•./•  replaces  x  and  l<:'"y  replaces  y, 
the  substitution  ,/■  =  r',  y  =  e"'h:.  wall  lower  the  degree  of  the  equation. 
It  may  be  recalled  that 

rr,/  =  r.("'-">'(/>^  +  ni)  (1),  +  /M  -  3)  •  •  •  {I),  +  m  -  n  +  I)'-.      (15") 


ADDITIONAL  ORDINARY  TYPES  237 


Consider  xyy"  —  xij"^  =  ijy'  +  bzy""/^d-  —  -r'-.  If  in  this  equation  y  be  replaced 
by  ky  so  tliat  y'  and  y"  are  also  replaced  by  ky'  and  ky'\  it  appears  that  the 
equation  is  merely  multiplied  by  Ic^  and  is  therefore  homogeneous  of  the  first 
sort  mentioned.    Substitute 

V  =  t^,         y'  =  c~z\         y"  =  c'{z"  +  z'-). 
Then  e-^  will  cancel  from  the  whole  e(iuation,  leaving  merely 

xdz'       1   ,  bxdx 


z    =  z  +  hxz^/^  a-  —  X-     or    — ax  — 


v/,,2 


The  equation  in  the  first  form  is  liernoulli  ;  in  the  second  form,  exact.    Then 
—  =  0  V«-  —  X-  +  (7     and     dz 


z'  b  Va-  —  XT  +  C 

The  variables  are  separated  for  the  last  integration  which  will  determine  z  =  log?/ 

as  a  function  of  x. 

Again  consider  x* — ^-  =  (x^  +  2xy)-'  —  4y'-.    If  x  be  replaced  by  kx  and  y  by 
dx-  dx 

k-y  so  that  y'  is  rejilaced  by  ky'  and  y"  remains  unchanged,  the  equation  is  nuilti- 

plied  by  k*  and  hence  comes  under  the  fourth  type  mentioned  above.    Substitute 

x  =  e',         y  =  e^'z,         Il-y  =  €'{!),  + 2)  z,         J);y  =  {Dt  +  2){D-,  +  l)z. 

Then  c*'  will  cancel  and  leave  z"  +  2  {1  —  z)  z'  =  0,  if  accents  denote  differentiation 
with  respect  to  t.  This  equation  lacks  the  independent  variable  t  and  is  reduced 
by  the  substitution  z"  =  z'dz'/dz.    Then 

d"^'  d^  dz 

—  +  2  (1  -  z)  =  0,        z'  =  -~  =  {\-  z)-  +  C,        =  dt. 

dz         ^         '  dt      ^         '  (1-2.2)  4-0' 

There  remains  only  to  perform  the  quadrature  and  replace  z  and  t  by  x  and  ?/. 
103.   If  the  equation  may  he  obtained  by  differentiation,  as 

/        (hi         (i"i/\     (/n     en     cQ   ,  dn     , ,     ,^ 

*(■"■'  •"•  i'  ■  ■  ■■  S?)  =  7a7  =  si-  +  a^ ■"  +  ■  ■  ■  +  li^^y'-''  (i«) 

it  is  called  an  exact  equation^  and  f2  (.'■,  //,  //',  •  •  •,  ;/^"~^^)  =  C  is  an  inte- 
gral of  ^  =  0.  Thus  in  ease  the  equation  is  exact,  the  order  may  be 
loAvered  by  unity.  It  may  l)e  noted  that  unless  the  degree  of  the  ?ith 
derivative  is  1  the  equation  cannot  be  exact.    Consider 

*(•'■,.'/,//',  ■■■,  //"^)  =  <^y"^  +  <^,, 

where  the  coefficient  of  //'^"^  is  collected  into  <f>^.  Now  integrate  cfi^,  par- 
tially regarding  only  i/'-"~^^  as  variable  so  that 

That  is,  the  expression  *  —  fi/  does  not  contain  //'•"''  and  may  contain 
no  derivative  of   order  hisj'her  than  n  —  l\  and   mav  be   collected   as 


J4>^fy"'-' 

^  =  ^, 

ex 

Then 

*- 

,l,n-h 

238  DIFFERENTIAL   EQTATIOXS 

indicated.  Now  if  ^  was  an  exact  derivative,  so  must  *  —  Q[  be.  Hence 
if  m  ^  1,  the  conclusion  is  that  ^  was  not  exact.  If  vi  =  1,  the  i)i'ocess 
of  integration  may  be  continued  t(j  obtain  O.,  by  integrating  partially 
with  respect  to  y(«-*-i).  And  so  on  until  it  is  shown  that  ^  is  not  exact 
or  until  *  is  seen  to  be  the  derivative  of  an  expression  f2^  -f  O.,  +  •  •  ■  =  C. 

As  an  example  consider  ^  —  x-y'"  +  sy"  +  (2  x?/  —  1)  y'  +  y-  —  0.    Then 
^1  =  j'''-'^y"  =  -'"y"^  ^  -  Oj'  =  -  xy"  +  (2  xy  -  1)  y'  +  2/', 

fio  =  f-  sdy'  =  -  xy',         <i^  -9.[-  9.!,  =  2  jyy'  +  r  =  (-^r)'- 
As  the  expression  of  the  tir.st  onU'r  is  an  exact  derivative,  the  result  is 

^  —  r>i  —  9/y  —  {xy-y  =  0 ;     and     >I'^  =  x"y"  —  xy'  +  xy-  —  C\  =  0 
is  the  new  equation.    The  method  may  be  tried  again. 

fij  =  i^ xMy'  =  x-y',         ^^-n[=  -oxy'  +  xy-  -  C\. 

This  is  not  an  exact  derivative  and  the  equation  ^I'j  =  0  is  not  exact.  Moreover 
the  equation  4'j  =  0  contains  both  x  and  y  and  is  not  homoireneous  of  any  type 
except  when  C'j  =  0.  It  therefore  appears  as  though  the  further  integration  of  the 
equation  4'  =  0  were  inq)nssible. 

The  method  is  a})plied  witli  es})ecial  ease  to  the  case  of 


'^0  (h:"  ^  ^ '  <i 


(1,1 


••.  +  A„_   -'-  + A,,// -/?(.■;=  0,  (17; 

(I.I 


where  the  coefficients  are  functions  of  ,/■  alone.  This  is  known  as  the 
linear  I'liiKition,  the  integi-ation  of  which  has  l)eeii  treatoil  onlv  when 
the  order  is  1  or  when  the  eoctficit'iits  are  constants.  The  apjilieation 
of  successive  integration  by  jnirts  gives 

fij  =  x,n("-'\   ^,=  (^\  -  ^'>u/""'\   ^,  =  (X  -  a;  +  A-,;';/«-^'\  ■  .  •  : 

and  after  ?i  such  integrations  there  is  left  merely 

(.Y„  -  A,:_,  +  •  ■  ■  +  r-l)"^^Aj  +(■-!  )"AV)//  -  /.', 
which  is  a  derivative  only  wlion  it  is  a  function  of  ./■.    Hence 

A„  -  A,:_i  +  . . .  +  f-  1  y'-K\\  +  (  -  ]  )" a;  =  0  (18) 

is  the  condition  that  tlie  linear  equation  shall  be  exact,  and 

^^c/"~'^  +  (-Vj  -  a;) //<"--)  +  CA,  -  A-;  +  Aj')y"-«>  +  •  ■  •  =  f/^/.'-  (19) 

is  the  first  solution  in  case  it  is  exact. 

As  an  example  take  //'"  +  ;/"cos./-  —  2//'sin.r  —  yco^x  =  sin2./-.    Tlie  test 
A'.;  —  a;'  +  A'j"  —  A',"'  =  —  cus  x-  +  2  cos.c  —  cos  x  =  0 


ADDITIONAL   ORDIXAEY   TYPES  239 

is  satisfied.  The  integral  is  therefore  y"  +  y' coax  —  yainx  =— lc(ii^2x  +  C\. 
This  equation  still  satisfies  the  test  for  exactness.  Hence  it  may  be  integrated 
again  -with  the  result  y'  +  y  cos  x  =  —  ^sin2j;  +  C\x  +  C„.  This  belongs  to  the 
linear  type.    The  final  result  is  therefore 


y 


■sin.r   rc^mx^C\X  +  C'j) 'iX  +  CgC-^*"'^  +   |  (1  —  slu  x). 


EXERCISES 

1.  Integrate  these  equations  or  at  least  reduce  them  to  (juadratures : 
(a)  -Ixir'n"  ^  y"-^  -  a"-,  ((3)   (1  +  x"-)  y"  +  1  +  Z/'^  =  0, 

(7)  Z/"'  +  "-y"  =0,  (5)  y^-  -  m-y"'  =  e%         (e)  x-y^''  +  a-y"  =  0, 

(i')   "-//"//'  =  -r.  iv)  -''y"  +  .'/'_=  •"',  ('9)   //'".'/"  =  -1, 

(0   (1  -  .r2) y"  -  xy'  =  2,  {k)  //^-  =:  ^  g'" ,  (X)  y"  =/(j/), 

(m)  2  (2  a  —  y)  y"  -  1  +  y'",  {v)  yy"  —  y'-  —  y"-y'  =  0, 

(o)  yy"  +  y'-  +1  =  0,  (tt)  2 y"  =  t",  (p)  y"y"  =  a. 

2.  Carry  the  integration  as  far  as  possible  in  these  cases: 
(a)  x~y"  -  {mx'y'2  +  ny^)-^,  {/3)  mx"y"  =  (y  —  xy')-, 

(y)  •r'y"  =  (y  -  ry'f,  (5)  x^y"  -  r'y'  -  .rV^  +  4  2/2=0, 

(e)  x--//"  +  x--i(/  =  i?/'2,  (f)  r/////"  +  }>y'-'-  -  yy'{c-  +  .r^)-  n. 

3.  Carrj'  the  integration  as  far  as  possilile  in  these  cases: 

(a)  ((/2  +  J-)  y'"  +  0  i//?/"  +  y"  +  2  //'-  =  0.  i'fi)  y'y"  -  ?/x-//'  =  xy2, 

(7)  x3(/i/'"  +  'Zx'^y'y"  +  O.riyy"  +  Oj--//'-  +  Wxyy'  +  3  */-  =  0, 

(5)  y-\-Zxy'  ^-2  yy'->  +  (./-^  +  2  //■^i/')  2/"  =  0, 

(6 )  (2  x""/  +  /-2/)  y"  +  4x-//'2  +  2  xyy'  =  0. 

4.  Treat  these  linear  equations: 

(a)  xy"  +  2  ?/  =  2  .r,  (/3)   (x2  -  1)  y"  +  4  xy'  +  2  ?/  =  2  x, 

( 7)  y"  -  y'  '■'  't  X  +  y  csr2  .r  =  cos  .r,  (5)   (,r'-  -  x)  y"  +  (?>  x  -  2)  y'  +  y  =  0, 
(e  )   (,r  —  x"')  y'"  +  (1  —  'j  X-)  y"  —  2  xy'  +  2  y  =  r.  x. 

( J-)   (X-'  +  x2  -  .3  X  +  1 )  y'"  +  (1»  X'-  +  0  X  -  !>)  y"  +  ( 18  x  +  6)  y'  +  (i  y  =  x'', 
(t?)   (x  +  2)--  y'"  +  (x  +  2)  y"  +  y'  =  1.        (0)  xV'  +  3xy'  +  y  =  x, 
( t )   {x-  -  X)  y'"  +  (8  x2  -  :|)  y"  +  14  xy'  +  4  y  =  0. 

5.  Note  that  Ex.  4  (0)  comes  under  the  tliird  homogeneous  type,  and  that  Ex.  4 
(77)  may  be  brought  under  that  type  by  nuUtiplying  by  (x  +  2).  Test  sundry  r)f  Exs. 
1,  2,  8  for  exactness.  Sliow  tliat  any  linear  e(iuati<in  in  which  the  coefficients  are 
polynomials  of  degree  less  than  the  order  of  the  derivatives  of  which  they  are  the 
coefficients,  is  surely  exact. 

6.  Sometimes,  when  the  condition  that  an  eijuation  lie  exact  is  not  satisliefl,  it 
is  possible  to  find  an  integrating  factor  for  the  eijuation  so  that  after  nudtiplication 
by  the  factor  the  equation  becomes  exact.    For  linear  equations  try  x'".    Integrate 

{a)  xhj"  +  (2  X*  -  X)  y'  _  (2  x^  -  1)  y  =  0.         (/3)   (x^  -  x^)  y"  -  xh/  -  2  y  =  0. 

7.  Show  that  the  equation  y"  +  Vy'  +  Qy'-  =  0  may  be  reduced  to  quadratures 
1^  when  V  and  Q  are  both  functions  of  y.  (ir  'P  when  both  are  functions  of  x,  or  ?P 
when  /'  is  a  function  of  x  and  Q  is  a  function  of  y  (integrating  factor  1/y').  In 
each  case  find  the  general  expression  for  y  in  terms  of  (juadratures.  Integrate 
y"  +  2  y'  cot  x  +  2  y'^  tan  y  -  0. 


240  DIFFERENTIAL  EQUATIONS 

8.  Find  and  discuss  the  curves  for  which  the  radius  of  curvature  is  proportional 
to  the  radius  r  of  the  curve. 

9.  If  the  radius  of  curvature  I\  is  expressed  as  a  function  R  =  7i(.s)  of  the  arc  s 
measured  from  some  jioint,  tlie  ecjuation  U  =  li{^)  or  s  =  •s(/^)  is  called  the  intrinsic 
equation  of  the  curve.  To  lind  the  relation  between  x  and  y  the  second  equation 
may  be  differentiated  as  cU  =  s'{Ii)dU,  and  this  equation  of  the  third  order  may  be 
solved.  Show  that  if  the  origin  be  taken  on  the  curve  at  the  point  s  =  0  and  if  the 
X-axis  be  tangent  to  the  curve,  the  equations 

X  =    I     cos     I    :  -    ds,         y  =    (    sin      |     —    ds 

express  the  curve  parametrically.    Find  the  curves  whose  intrinsic  equations  are 
{a)  R  =  a,         (/3)  aR  =  .s^  +  ry2^         (7)  7.'2  +  ,s2  =  10  ^(2_ 

10.  Given  F  =  y('>)  +  .Y,?/("-i)  +  X.,y("--^)  +  •  •  •  +  ^'u  -^1!/'  +  ^V„y  =  0.  S)  ow 
that  if  ^i,  a  function  of  x  alone,  is  an  integrating  factor  of  the  equation,  tlien 

*  =  fx(")  -  (-V)("-i)  +  (A»("---) +  (_  1)"-i(.Y„_im)'  +  (-  1)«A'„M  =  0 

is  the  equation  .satisfied  by  fi.  Collect  the  coeilicient  of  n  to  show  that  the  condition 
that  the  given  eiiuation  be  exact  is  the  condition  that  tins  coefficient  vanish.  The 
equation  <!>  =  0  is  called  the  adjoint  of  the  given  equation  F  =  0.  Any  integral  /x 
of  the  adjoint  etjuation  is  an  integrating  factor  of  tlie  original  e(iuation.  Moreover 
note  that 

J^pLFdx  =  /xyO'-i)  4.  {fMX\-  ^')y (>'--)  +...  +  (_  1)"  J y<t>dx, 

or  d[pLF  -  (-  l)"^*]  =  d  [txy("  -D  +  {,xA\  -  m')  y("  -■;)+...]=  dU. 

Hence  if  /xF  is  an  exact  differential,  so  is  y<i>.  In  other  words,  any  .solution  y  of  the 
original  ecjuation  is  an  integrating  factor  f(ir  thi'  adjoint  e(iuation. 

104.  Linear  differential  equations.    The  equations 

-V,/>"//  +  -\\/>"-\'/  +  ■■■+  X„  _,  !>;,  +  A,,//  =  R  (.>■), 

A\D";/  +  A^/>"  -\y  +  •  •  •  +  A„  _,J>^  +  A„y  =  0  ^-   ^ 

are  linear  differential  e(|uations  of  tlie  ??tli  order;  the  first  is  called  the 
ro//q/li'f(',  cf/itiifi'Di  and  the  seeoiul  the  rcliiri'd  i-fitnifhni.  If  y^,  _y.,,  y/^,  •  •  • 
are  any  solutions  of  tlie  reduced  e(|uation,  and  C^,  (',,,  (\^,  ••■  are  any 
constants,  then  y  =  r '^y^ -(- r ',//_ -|- r '^y^ -|- . . .  is  also  a  solution  of  tlie 
I'cduced  equation.  This  follows  at  once  from  the  linearity  of  the  reduced 
equation  and  is  proved  by  direct  substitution.  Furthermore  if  /  is  atiy 
solution  of  the  complete  ecjuation,  then  //  +  /  is  also  a  solution  of  the 
com])lete  equation  (cf.  §  90). 

As  the  equations  (20)  are  of  the  ??t]i  order,  thi'v  will  detei'inine  //<'"' 
and,  by  differentiation,  all  lii,n-her  derivatives  in  terms  of  tlie  values  of 
■''■> ,'/; .'/')  ■  ■  •  5  .'/'""'"'•  1  lence  if  the  values  of  the  n  quantiries  //^,  //', ,  ■  •  • .  ,'/!■"  ^'' 
wdiich  coi'res})on(l  to  tlie  value  .r  =  .r^^  be  g-iven,  all  the  higher  derivatives 
are  detei'inined  (§^  <S7-.SS).  Ilciiee  there  are  71  and  no  more  than  /iarlii- 
trary  conditions  that  may  be  imposed  as  initial  conditions.    A  solution 


ADDITION^AL  ORDINARY   TYPES  241 

of  the  equations  (20)  which  contains  n  distinct  arbitrary  constants  is 
called  the  general  solution.    By  distinct  is  meant  that  the  constants 
can  actually  be  determined  to  suit  the  n  initial  conditions. 
If  y^  y.,,  •  •  •,  y„  are  n  solutions  of  the  reduced  equation,  and 

y  =  ^\Ui        +  ''  'J'i        -^ ^  '""•'/« ' 

U'  =  f'l'A        +  ''iU'i        -I 1-  ^-\^u'n^  (21) 

y(«-i)  ^  Ci//^"-^>  +  cv/^"-'^  H 1-  c„//,^;'-^>, 

then  y  is  a  solution  and  y',  ■  ■  ■ ,  //'"  ~^^  are  its  first  ?i  —  1  dei'ivatives.  If 
.r^  be  substituted  on  the  right  and  the  assumed  corresponding  initial 
values  y^,  ii^,,  ■■  ■ ,  y["~^^  l)e  sul)stituted  on  the  left,  the  above  n  equations 
l^ecome  linear  equations  in  the  n  unknowns  (\,  C\,  ■  ■  ■ ,  C\, ;  and  if  they 
are  to  be  soluble  for  the  C's,  the  condition 


^''(z/p  2/2'  •••'  Vn) 


ii\        111        ■■■    y'n 


//I"-"  ti^'-''  ■■■  '/:: 


^  0  (22) 


must  hold  for  every  value  of  r  =  ,/;..  Conversely  if  the  condition  does 
hold,  the  equations  will  be  soluldc  for  the  '"'s. 

The  determinant  M'i;/^,  //.,,  •••,  //„)  is  called  the  Wrons]:i>in  of  the  n 
functions  y^,  //.,,  ••-,  -/„.  The  result  may  be  stated  as:  If  n  functions 
Uii  Vti  ■  ■  ■)  ]ln  '^vhich  are  solutions  of  the  reduced  e(|uation,  and  of  which 
the  Wronskian  does  not  vanisli,  can  be  found,  the  general  solution  of  the 
reduced  equation  can  be  Avritten  down.  In  general  no  solution  of  the 
equation  can  ])e  found,  whether  by  a  detinite  jirocess  oi'  by  inspection; 
but  in  the  rare  instances  in  whicli  the  n  solutions  can  l)e  seen  by  inspec- 
tion the  problem  of  the  solution  of  the  reduced  equation  is  completed. 
Frequently  one  solution  may  V)e  found  by  inspection,  and  it  is  therefore 
important  to  see  how  much  this  contributes  toward  effecting  the  solution. 

If  y^  is  a  solution  of  the  reduced  equation,  make  the  substitution 
y  =  y^z.  The  derivatives  of  //  may  be  obtained  by  Leilniiz's  Theorem 
(§  8).  As  the  formula  is  lijiear  in  the  derivatives  of  re,  it  follows  that 
the  result  of  the  sul)stitution  will  leave  the  equation  linear  in  the  new 
variable  z.  ^Moreover,  to  collect  the  coefficient  of  z  itself,  it  is  necessary 
to  take  only  the  first  term  y[^"'z  in  the  expansions  for  the  derivative  y'^^'K 

^«^C«  (A;,./^)  +  X,y["  -')+...+  A-„  _,y[  +  A',./,)  ..  =  0 

is  the  coefficient  of  z  and  vanishes  In-  the  assumption  that  y^  is  a  solu- 
tion of  the  reduced  equation.    Tlien  tlie  equation  for  z  is 

P/'"  +  P/"  -^>  +  ■  ■  •  +  Pn-,^"  +  Pu-i-'  =  0 ;  (23) 


242  DIFFEREXTIAL  EQUATIONS 

and  if  z'  be  taken  as  the  variable,  the  equation  is  of  the  order  71  —  1. 
It  therefore  appears  that  the  kiiowleilge  of  a  solution  //^  ri'(hi.C(;s  the  order 
of  the.  equation  hy  one. 

Now  if  y,,  V-^t  ■  • '  ■)  Vp  were  other  solutions,  the  derived  ratios 

0,     .^(S.     -,     .-.  =  ©■       (-) 

would  be  solutions  of  the  equation  in  z^  \  for  l)y  substitution, 

y  =  I'x^x  =  yv      y  =  y^i  =  y^^      ■  •  • '      y  =  y^p  -i  =  %, 

are  all  solutions  of  the  equation  in  y.  ^Moreover,  if  there  were  a  linear 
relation  <\z\  -|-  '''0.-2  +  •  •  •  4-  C'^, _r^p_i  =  0  connecting  the  solutions  .v-, 
an  integration  would  give  a  linear  relation 

C'l//.,  +  C.jj^  +  •  •  •  +  C;_i.y„  +  C^y^  =  0 

connecting  the  7?  solutions  y;.  Hence  if  there  is  no  linear  relation  (of 
which  the  coefficients  are  not  all  zero)  connecting  the  ^^^  solutions  ?/,■  of 
the  original  equation,  there  can  be  none  connecting  the  y-'  —  1  solutions 
z\  of  the  transformed  equation.  Hence  a  Invnrledge  of  p  soluti<ms  of 
the  original  n^duced  equation  gives  a,  new  redueed  equation  of  xrhich 
p  —  1  solutions  are  knoicn.  And  the  process  of  substitution  may  be 
continued  to  reduce  the  order  further  until  the  order  n  —  p  is  reached. 

As  an  example  consider  the  equation  of  tlie  third  order 

(1  -  X)  y'"  +  (./••-  -\)y"  -  s-y'^  sy  =  0. 

Hero  a  simple  trial  shows  that  x  and  e^  are  two  solutions.    Substitute 

y-c'z,       y'  =  e'-{z  +  z'),       y"  =  c^{z +  2z'  +  z"),       y'"  =  c-'iz  +  oz' +  3z"  +  z'"). 

Then  (1  -  x)  z'"  +  [s"  -  3  x  +  2)  z"  +  (x-  -  3  x  +  1)  z'  =  0 

is  of  the  second  order  in  z' .    A  known  solution  is  the  derived  ratin  (x/e'")'. 

z'  =  (xt-  ■'■)'  =  e-  ■'■  (1  —  x) .    Let  z'  =  C"  -^  (1  —  x) ;';. 

From  this,  z"  and  z'"  may  be  found  and  the  ecjuation  takes  tlie  form 

'///•'  2 

(1  —  x)  ir"  +  (1  +  x)  (x  —  2)  ;'•'  =  0     or     -  --  =;  X'Zx ds.  ■ 

u:'  X  —  1 

This  is  a  linear  (Mjuation  of  the  first  f)rder  and  may  be  solved. 

log  v:'  ^  I  X-  -  2  log  (x  -1)4-  C     or     1//  -  C^t^-^'ix  -  1)--. 


Hence 


='=©'"=^.(.')7'^"<'-'>-'"^+^=(fJ' 

y  =  c^z  =C\r'   fi^''-  )    fr-  '\.r  -  })~-(,lr)-  +  C.-,X  +  C.C''. 


ADDITIOXAL  ORDIXAEY  TYPES  243 

The  value  for  y  is  tlius  obtained  in  terms  of  quadratures.  It  may  be  shown  that  in 
ease  tlie  equation  is  of  the  >ith  degree  with  j)  known  solutions,  the  final  result  will 
eall  forj)()i  — p)  quadratures. 

105.  If  the  general  solution  y  =  6'^//^  + '''.,.'/.,  +  •  ■  ■  +  t'„y„  of  the  redticed 
equation  has  been  found  (called  tlie  cnmphnnentanj  functlnn  for  the 
complete  equation),  the  general  solution  of  the  complete  equation  may 
alwavs  be  ol)tained  in  terms  of  quadratures  by  the  important  and  far- 
reaching  iiicihiiil  nf  tJin  vdr'utfuni  of  ronstants  dtie  to  Lagrange.  The 
question  is  :   ( "unnot  functions  of  :r  be  found  so  that  the  expression 

!/  =  Cp-)  II,  +  <-  •,(.'■ )  //,  +  •  •  ■  +  Cj.r)  //„  (24) 

shall  be  the  solution  of  the  complete  eijuation  '.'  As  there  are  n  of  these 
functions  to  be  determined,  it  should  l)e  })Ossible  to  impose  n  —  1  condi- 
tions upon  them  and  still  find  the  functions. 

Differentiate  //  on  the  supposition  that  the  f's  are  variable. 

u'  =  < \'j\  +  <^'^ij'i  -f-  •  •  •  +  ('n'L  +  ih^'\  +  i// ■;  +  ■■■  +  y,S'\. 

As  one  of  the  conditions  on  the  '"''s  su})pose  that 

y/'; +  y/'; +  ••■  +  //,/■;  =  0- 

Differentiate  again  and  impose  tlie  new  condition 

ll\<"x  +  !l'i'"-i  +  ---  +  !l'j"n=^, 
«o  that  y"  =  f\y';  +  rj;  -f-  •  •  ■  +  ^  '„//:;. 

The  dilferentiation  may  be  continiK^l  to  the  (a  —  1  )st  condition 

ii\"  --'''\  +  ii-r  -  '*^  ■;  +  •••  + '/::  -  ''^':  =  o, 

and  7/^"  -' '  =  (\y'{  -^ '  +  ^  ',//.V'  ^"4 V( '  „y\:  -^\ 

Then  //^»>  =  (  \y\'>  +  (\y^''  +■■■  +  cJ:' 

Now  if  the  expressions  thus  found  for  //,  //'.  //",  •••,  //^""'\  //^"^  be 
sul)stituted  in  the  conq>lete  e(|Uatiou,  and  it  lie  remendjcred  that  y^, 
.'/..•  ■  ■  ■  ;  .'/,,  ''^'P  solutions  of  the  reduced  (Mpuition  and  lieiice  givt;  0  Avhen 
substituted  in  the  left-hand  side  of  the  e(juation,  tlie  result  is 

M""''^';  4-  llT""'''-!  4-  •  •  •  4-  y\r'''"..  =  li- 
Hence,  in  all,  there  are  n  linear  equations 

y/"i         +  lli'-"-i         +■■■  +  !l,Pn        =  0, 

y\''\      +u'/''i      +---  +  ii'/"u      =0, 


(25) 


244  DIFFERENTIAL  EQUATIONS 

connecting  the  derivatives  of  the  ("*s  ;  and  these  may  actually  be  solved 
for  those  derivatives  "which  will  tlien  be  expressed  in  terms  of  x.  The 
C's  may  then  be  found  by  (juadrature. 

As  an  example  consider  the  equation  witli  constant  coefficients 

(W  -\-  I))i/  =  sec  X     witli     y  =  C\  +  C,  cos  x  +  C^  sin  x 

as  the  sohition  of  the  reduced  equation.  Here  tlie  solutions  y^,  ?/,-,,  y^  may  be  taken 
as  1,  cosx,  sinx  respectively.  The  conditions  on  the  derivatives  of  the  C"s  become 
by  direct  substitution  in  (25) 

C'l  +  cosxt'.^  +  sin.f(''j  =  0,   —  i^mxC',  +  cosxf'^  =  0,   —  cosjCo  —  sinxC'g  =  secx. 

Hence  C[  =  sec  x,         r,^  =  —  1,         C'.^  =  —  tan  x 

and     C\  =  log  tan  (.^  x  +  i  tt)  +  r^ ,  (',,=  —  x  +  c, ,  C'g  =  log  cos x  +  c.,. 

Hence       y  =  c^  +  log  tan  (J  x  +  \  tt)  +  ('■.,  —  x)  cosx  +  (r^  +  log  c<js  x)  sin  x 

is  the  general  solution  of  the  complete  e(]uation.  This  result  could  not  be  obtained 
by  any  of  the  real  short  methods  of  >;;§  HO-UT.  It  could  be  obtained  by  the  general 
method  of  §  95,  but  with  little  if  any  advantage  over  the  method  of  variation  of 
constants  here  given.  The  present  method  is  equally  available  for  equations  with 
varial)le  coefficients. 

106.  Linear  ciiudtunis  of  ilti>  scrond  onJcr  are  especially  frequent  in 
practical  problems.  In  a  number  of  cases  the  solution  may  be  found. 
Thus  1°  when  the  coefficients  are  constant  or  nuiy  be  made  constant  by 
a  change  of  variable  as  in  Ex.  7,  }>.  222,  the  general  solution  of  the 
reduced  equation  nuiy  be  writtiMi  down  at  once.  The  solution  of  the 
complete  equation  may  then  be  fouml  by  t)btaining  a  ])arti(;ular  integral 
/  by  the  methods  of  §)?  95-97  or  by  the  ai)plication  of  the  method  of 
variation  of  constants.  And  2°  when  the  eipiation  is  exact,  the  solution 
may  ])e  had  by  integrating  the  liiu'ar  equation  (19)  of  §  103  of  the  first 
order  by  the  ordinary  metliods.  And  ?°  when  one  solution  of  the  re- 
duced equation  is  known  (§  104),  the  reduced  equaticjn  may  be  eom- 
])letely  solved  and  the  complete  e(ptation  nuiy  then  be  solved  by  the 
method  of  variation  of  constants,  or  tlie  complete  equation  may  be 
solved  directly  by   Ex.  O  below. 

Otherwise,  AVi'ite  tlin  differential  ('(juation  in  the  form 

(p-ii  (III  .^_. 

7-,  +  /'  /  +  (III  =  11.  (2G) 

dx'  (IX 

The  substitution  //  =  i/z  gives  the  lU'W^  equation 

(Pz        rid  (I  \dr:       1       „  ,  R 

-^  +    -7-  +  P)  ~,  -  +  -  (/'"  +  P"'  +  Q")r.  =  -■  (2G') 

(/.I-         \  II  11 X  I  il.r         II  II 

If  II  l)e  determiutM]  so  that  the  coefficient  of  ,-;'  vanisht-s,  tlien 

ii  =  e-ll"'-     and     '!^^(u-y^;^-\Ar:  =  nAJ-^'^.      (27) 
f/.'"       \  Ji  dx        4       / 


ADDITIONAL  ORDINARY  TYPES  245 

Now  4°  if  Q  —  I  P'  —  I  P'  is  constant,  the  new  reduced  equation  in 
(27)  may  be  integrated ;  and  5°  if  it  is  A'/^"',  the  equation  may  also  be 
integrated  by  the  method  of  Ex.  7,  p.  222.  The  integral  of  the  com- 
plete equation  may  then  be  found.  (In  other  cases  this  method  may 
be  useful  in  that  the  equation  is  reduced  to  a  simpler  form  where  solu- 
tions of  the  reduced  equation  are  moTe  evident.) 

Again,  suppose  that  the  independent  variable  is  changed  to  z.    Then 

ilZ  t^  ((.V  .V  ^ 

Now  6°  if  s'"^  =  ±(l  will  make  .-s"  -|-  Pz'  =  hz'-,  so  that  the  coefficient 
of  dy/dz  becomes  a  constant  /.-,  the  equation  is  integrable.  (Trying  if 
z'-  =  ±  Qz-  will  make  ,~"  +  Pz'  =  l:z'- jz  is  needless  because  nothing  in 
addition  to  6°  is  thereby  obtained.  It  may  happen  that  if  z  be  deter- 
mined so  as  to  make  ,t"  +  P^'  =  0,  the  equation  will  Ije  so  far  simpli- 
fied that  a  solution  of  the  reduced  equation  becomes  evident.) 

^      .  1        ,  ,     fl'V      2  dy      «-         ^     ^T  ,     .       . 

Con.siaer  the  example 1 1 ?/  =  0.     Here   no   .sohition   is   apparent. 

Hence  compute  Q  —  i  -P'  —  5  P'-  Thi.s  is  a^/x'^  and  i.s  neither  constant  nor  propor- 
tional to  l/x^.  Hence  the  methods  4°  and  5°  will  not  work.  From  z'^  =  Q  =  a'^/x* 
or  z'  =  a/x",  it  appears  that  z"  +  Pz'  =  0,  and  G"^  works  ;  the  new  equation  is 

-^  4-  ?/  =  0     with     z  = 

dz-  X 

The  .solution  is  therefore  .«;een  immediately  to  be 

y  =  C,  cosz  —  C,  sin  2     or     ?/  =  C\  coi^{a/x)  -f-  C^  sin  Ut/x). 

If  there  had  been  a  riirht-hand  meinbcr  in  the  oriirinal  ecjuation,  the  .solution  could 
have  been  found  by  the  method  of  variation  of  constants,  or  by  some  of  the  short 
methods  for  finding  a  particular  .solution  if  /*  had  been  of  the  proper  form. 

EXERCISES 

1.  If  a  relation  C■^y■^  +  C.,//.,  +  •••-!-  C„y„  =  0,  with  constant  coefficients  not  all  0, 
exists  between  n  functions  ?/,,?/.,.•••.  ij„.  of  x  for  all  values  of  x.  the  functions  are 
by  definition  said  to  be  linearly  dependent;  if  no  such  relation  exists,  they  are  said 
to  be  linearly  independent.  Show  that  the  nonvaiiLshing  of  the  Wronskian  is  a 
criterion  for  linear  indepemlence. 

2.  If  the  general  .solution  y  —  C\y^  +  C.,y.,  +  ■  ■  •  4-  C„y„  is  the  .same  for 
X^yM  +  A\y(n -1)  +  .  .  .  +  X.jj  =  0     and     P,^yO»  +  P^yC -!)  +  ...  +  P„y  =  0, 

two  linear  equations  of  the  nth  order,  show  that  y  .satisfies  the  equation 

{A\P,  -  A\^P,)  y(n  -1)  +  .  .  .  +  {X„P,  -  XJ\)  y  =  0 

of  the  (n  —  l)st  order;  and  hence  infer,  from  the  fact  that  y  contains  n  arbitrary 
constants  corresponding  to  n  arbitrary  initial  conditions,  the  important  theorem: 
If  two  linear  equations  of  the  7(tli  urder  have  the  same  general  solution,  the  corre- 
sponding coefficients  are  proportional. 


246  DIFFERENTIAL  EQUATIONS 

3.  If  2/j ,  2/2,  •  •  • ,  2/n  ivi'e  n  independent  solutions  of  an  equation  of  tlie  nth  order, 
show  that  the  equation  may  be  taken  in  the  form  ^V{y^,  y.^,  ■  ■  ■ ,  Pn,  y)  =  0. 

4.  Sliow  that  if,  in  any  reduced  equation,  A',j_i  +  ^A',,  =  0  identically,  then  x 
is  a  solution.   Find  the  condition  that  x'"  be  a  solution ;  also  that  e""^  be  a  solution. 

5.  Find  by  inspection  one  or  more  independent  solutions  and  integrate  : 
(a)  {l  +  x"-)y''-2xy'  +  2y  =  0,  (^)  xr/'  +  {I- x)y' -  y  =  0, 

(7)  {ax  -  bx^-) y"  -  ay'  +  2hy  =  0,  ( 5 )  i  ?/"  +  xy'  -  {x  +  2)y  =  0, 

(e)   (logx  +  ^  -  4  +  -)y'"  +  i^ogx  +  1  +  i  -  iVr  +  (~  -  -){y'-xy)  =  0, 
\  X*       X-       X/  \  X-*       X"       x-/  \x-       x/ 

( i)  y'"' -  ■((/"+  xy'-  y  =  0,  (77)  (4 x'-^  -  x  +  1) y"'+  8 xry"-  4 xy'-  8 ?/  =  0. 

6.  If  ?/j  is  a  known  solution  of  the  equation  y"  +  Py'  -\-  Qy  =  I!  of  the  second 
order,  show  that  the  general  solution  may  be  written  as 

y  =  G^y^  +  C.^/i  \  e  J        —  +  yi—eJ  y^eJ        K{dx)^. 

'^  2/f       ''  y\  '^ 

7.  Integrate : 

(a-)  xy"-  (2 X  +  1) ?/'  +  (x  ■\-l)y-x^  —  x  —  1, 

(i3)  ?/'  -  xV  +  x?/  =  X,  (7)  X2/"  -{■  {I-  x)y'  -y  =  e% 

( 5 )  ?/"  —  xy'  +  {x  —  l)y  =  R,  {^)  y" si'i" -c  +  y' sin -c  cos x  —  y  =  x  —  sin x. 

8.  After  writing  down  the  integral  of  the  reduced  equation  by  inspection,  apply 
the  method  of  the  variation  of  constants  to  these  equations  : 

(a)  {Ifi  +  1)  2/  =  tan  X,  (/3)  (i>^  +  1)  y  =  see^  x,  (7)  {B  -  Yfy  =  e-'-(l  -  x)-  ^, 
(5)   (1  -  x)y"  ^xy'  -y^{\-  x)-\    (e)  (1-  2x  +  x^){y"' -  I)- xhj"  ^2x'y' -  y  =  1. 

9.  Integrate  the  following  equations  of  the  second  order: 

(or)  4  xhj"  +  4  x^?/'  +  (x^  +  lyhj  =  0,  (/3)  ?/"  -  2  y'  tan  x  -  (flS  +  i)  ^^  =  0. 

{7)  xy"  +  2y'  —  xy  =  2e'-',  (5)  ?/"sinx  +  2  ;/' cosx  +  3  ^sinx  =  e*-, 

(e )  y"  +  y'  tan  x  +  ^  cos^  x  =  0,  (f )  (1  -  x-)  y"  -  x;/'  +  -1  y  =  0, 

(7,)  2/"  +  (2  e^  -1)2/'  +  e2^-2/  =  £-4^,  [6)  xhj"  +  3  x^/  +  y  =  x-  2. 

10.  Show  that  if  ^^yy"  +  A'j?/'  +  X.^y  =  11  may  be  written  in  factors  as 

(A-,I;2  +  Xj  J^  +  A'2)  2/  =  {p^U  +  r/i)  (p,7J  +  r/,)  2/  =  K, 

where  the  factors  are  not  connnutative  inasmuuli  as  the  differentiation  in  one 
factor  is  applied  to  tlie  variable  coefticients  of  the  succeeding  factor  as  well  as 
to  i*,  then  the  solution  is  obtainable  in  terms  of  (quadratures.    Show  that 

(ilV-2  +  Vllh  +  l\'l-2  =  ^i      fi"'^      fJi'h  +  Pi'i-i  =  ^'2  ■ 
In  this  manner  integrate  tlie  f(jllowing  equations,  clioosing  p^  and  p.,  as  factors  of 
A'l^  and  determining  r/j  and  f/„  by  inspection  or  by  assuming  them  in  some  form  and 
applying  the  method  of  undetermined  coerticicuts  : 

(a)  xy"  +  (1  -  X)  y'-y  =  e-'\  (li)  3  x'^y"  +  (2  -  0  x^)  y'  _  4  =  0, 

(7)  Sx:^y"+{2  +  (ix-(]x^)y'-4y  =  0,  (5)  (x^- ]  )y"- (3x  +  1)  2/- x  (x -1)2/ =  0, 
( e )  axy"  +  (3  a  +  bx)  y'  +  3  by  =  0.  ( f )  xy"  -  2x  (1  +  x)y' +  2(1  +  x)  y  =  xK 

11.  Integrate  these  e(iuation.s  in  any  manner  : 

"I  r  4-  "\  r  8  *^  /  *^  \ 

{a)  y"  -  -  _i/  +  ■ '- 2/  =  0,  (/3)  y"  -  "^  y'  +    a''  +  -,    2/  =  0, 

Vx  ■*■'-■"  2;  \         x-f 


ADDITIOXAL  ORDINARY   TYPES  247 

(7)  2/"  +  2/' tan X  +  ?/ cos- X  =  0,  (5)  2/"-2(n  — "j  ?/'+ (  h--2— j?/  =  e'«, 

{ e)   (1  -  x2)  y"  -  xy'  -  c^y  =  0,  (f )   (a^  -  x^)  y"  -  8  x;/'  -  12  ?/  =  0, 

/    X     ..  1  /2      ,        \  ,^.     ,,      9-4x    ,      6-3x 

{'7)2/    +  -V-, 2/ =  e^-  -  + logx   ,  {&)  V    - —, 2/+^, 2/ =  0, 

x-logx  \x  /  3  — X  3  — X 

(0  y"  +  2  x-i;/'  -  n-y  =  0,  (k)  ?/"  -  4 x^/'  +  (4  .s^  -  3)  y  =  e^-, 

(\)  y"  +  2  n^/'  cot  nx  +  (//t-  —  )i-^)  ?/  =  0,         (/i)  ;/"  +  ^  (x-i  +  i'x--) ;/'  +  Ax-'^y  =  0. 

12.  If  y^  and  2/,  are  solutions  of  y"  +  P?/'  +  7?  =  0,  show  by  eliminating  Q  and 
integrating  tliat  - 

ViVi  -  2/o2/i  =  Ce  J 
What  if  C  =  0  ?  If  C  7i  0,  note  that  y^  and  ?/j  cannot  vanish  together  ;  and  if 
2/j(a)  =  y-^{h)  =  0,  use  the  relation  (y<,y[)a  ■■  (y.^y[)),  =  k>0  to  show  that  as  y\^^  and 
2/jj,  have  opposite  signs,  2/0 „  and  1/26  have  opposite  signs  and  hence  y.^i^)  =  0  wliere 
a  <^<b.  Hence  the  theorem  :  Between  any  two  roots  of  a  solution  of  an  equation 
of  the  second  order  there  is  one  root  of  eveiy  solution  independent  of  the  given 
solution.    What  conditions  of  continuity  for  y  and  y'  are  tacitly  assumed  here  ? 

107.  The  cylinder  functions.  Suppose  that  CJ-'')  is  a  function  of  x 
which  is  different  for  different  values  of  ?i  and  Avhicli  satisfies  the  two 
equations 

Cn -:(-^)  -  ^'n  +i(-x-)  =  2  -  C„{x),     C„ _,{x)  +  r'„  ^,(,, )  =  —  Cix).      (29) 

Such  a  function  is  called  a  cylinder  function  and  the  index  n  is  called 
the  order  of  the  function  and  may  have  any  real  value.  The  tw^o  equa- 
tioiis  are  supposed  to  hold  for  all  values  of  n  and  for  all  values  of  x. 
They  do  not  completely  determine  tlie  functions  but  from  them  follow 
the  chief  rules  of  operation  with  the  functions.  For  instance,  by  addi- 
tion and  subtraction, 

c:C'-)  =  r;_,(.>-)  -  ^  cjx)  =  -  c„(^)  -  c„^,(:x).  (30) 

Other  relations  which  are  easily  deduced  are 

j:y{x"C„(ax)^  =  a.r"r\^_^(a,:),  ]),[..-"(' Ja,')^  =  -  ax-C„^,(x),   (31) 

D^b^cXV^)]  =  h  V7u^C,,_,{V^:),  (32) 

C';(^)  =  -  ^^(•'•),  C_Jx)  =  (-  1)"CJ,'),         n  integral,  (33) 

C,Xx)K(x)  -  C:(x)K„(x)  =  C„^,(x)K\(x)  -  Cjx)K,,+,(,c)  =  ^,  (34) 

where  C  and  A'  denote  any  two  cylinder  functions. 

The  proof  of  these  relations  is  simple,  but  will  be  given  to  show  the  use  of  (29). 
In  the  first  case  differentiate  directly  and  substitute  from  (29). 


-Dx[x"G'„(ax)]  =x" 


alJaj-CJax)  +  -  Cn{ax) 

aC„-i{ax)  —  a —  C„{ax)  +  -  CJax)    . 
ax  -^  J 


248  DIFFERENTIAL   EQUATIONS 

The  second  of  (31)  is  proved  similarly.    Fur  (32),  differentiate. 

"  1       1—1  "  1        ~~  

JjJx-CJVT^)]  =  -  lu-      C'„(V(j-j-)+  X-      ^  "  D    ~~r„(Vax) 
2  2    \  J^     ^  '■ ' 


-,         _    "-In  -, 

^  Wax       '  ~  Vax  J 


Next  (33)  is  obtained  P  by  substituting;-  0  for  n  in  botli  e(iuations  (2it). 

r_,(.c)  -  C\{x)  =  2  r;{x).      r_i(,,')  +  r,(.,-)  =  O.     li.^nee      C^ix)  =  -  r',(.r)  ; 

and  2°  by  substitutinu'  successive  values  for  n  in  the  second  of  (21t)  written  in  the 
form  x(',i-]  +  .('('„  1^1  =  2  /(("„.    Then 

/C'_i  +  ^<'-'i  =  0.         x('^.2  +  f^'o^  --^('-1.         x(\^  +  xC.r=2C\, 
x('-3  +  x('_i  =  -  4  r_.,.  ./■(",  +,K',.  =  4  r.,. 

xC-i  +  .fr_._,  =  -  ()  r,-,.  xC.-,  +  x(\  =  i;  (".,. 

and  sn  on.  The  first  ^ives  ('_]  =  —  (\.  Subtract  the  next  two  and  u.se  C-i  +  ('^  =  0. 
Then  C'_2  —  (^'o  =  0  "i' ''-■.'  =  (— 1  )-^ '...•  Aild  the  next  two  and  use  the  relations 
already  found.  Then  /'_o  -f-<',.  —  0  <>v  C^g  =  (—  1  )•'•''.,.  Subtract  the  next  two. 
and  so  on.  For  the  last  of  the  relations,  a  very  important  one.  note  tirst  that  the 
t\V(j  expressions  become  eijui\alent  by  virtue  of  (2'.')  :   for 

C •„  7i';,  -  ( ■;,  A'„  -^  "  ( ■„  A",,  -  ( ■„  K„  X 1  -  "  C  ■„  K„  +  r„  :.  1 7v'„ . 
Now  ^lx{C„+^K„  -  C',;7v'„+,)J  -.  <■„  '.-xK„  -  <'..K„  .,  +  .rK,.i<'.,  -  "  "^  ^  ("„ 


+  xc,. .,(^ ;r„  - K„ .\ - r/r,, ,,Q  r'„  _  r„ ,, 


—  ,/•(„    7v  ,^ 7i  „  _u  1 

\  .'• 

Hence  .r  (r'„_,_i7v'„  —  f'ij\„  ;-i)  =  const.  =.1.  and  the  relation  is  jiroved. 

The  (■\liii(ler  functions  of  ii  o'ivcii  order  //  satisfy  a  linear  ditTerential 
('(jiiation  (d'  the  second  order.  Tliis  may  be  olitained  by  tiifbn'entiating 
tln^  lirst  (jf  ('1\))  and  eombiniiiL;-  willi  (."id). 


-  1 


^•„-i-^^\,+ 


'^^K. 


"  1 


irenee  •',  +       /    +    1-     .,)'/  =  0. 


.'/  =  ''„(■'■)■ 


C^o) 


This  ('(luation  is  known  as  Ilr.^sr/'s  n/i/t/finn  .■  llie  functions  '"„('.'•).  wlncli 
liave  been  called  cyliiulcr  functions,  are  often  called  /Irssr/'s  j'l/ //<■/)',, us. 
Fr<jui  the  etiuation  it  follows  that  an\  three  fumdions  of  the  same  onlci' 
/I  are  connected  li\'  a  liiu'ar  I'clation  and  there  are  only  {wo  indejiendeut 
fiimdions  ot  an\'  ljImmi  (jnler. 


ADDITIONAL   ORDIXARY   TYPES  249 

By  a  change  of  the  independent  variable,  the  IJessel  equation  may 
take  on  several  other  forms.  The  easiest  way  to  Ihid  them  is  to  operate 
directly  with  the  relations  (31),  (32).    Thus 

=  -  •'-""'^'.+1  +  2(m  +  l).t-"-V'„^j  -  x-"r„. 

Hence         '"^  +  <  ^+  "-''  ^  '^^- +  ,  =  0,  y  =  :.-  "CJ,).  (3G) 

(1.1-  X  a.r  '  ^      ^ 

Again  ?p^^+«-^l^U^  +  ^=0,         :,/  =  ,..■(,,■).  (37) 

dx  X  dx 

Also  xy"  -f  (1  +  n)  //'  +  //  =  0,  //  =  ./■"' - ( '„( 2  V.7).  (38) 

And  .'/•//"  +  (1  -  n)  ii'  +  11  =  0,         y  =  .>^  <  '„(  2  V^' ).  (39) 

In  all  these  differential  equations  it  is  well  to  restrict  ./■  to  positive  values 

inasmucli  as,  if  7i  is  not  specialized,  the  })Owers  of  x,  as  ./■",  ./•"  ",  x',  x  '"',  are 
not  always  real. 

108.  The  fact  that  ?i  occiirs  only  squared  in  (35)  shows  that  both 
Cj.r)  and  C_Jx)  are  solutions,  S(j  that  if  these  functions  are  inde- 
])endent,  the  conq)lete  solution  is  y  =  aC\^  +  ^>''-n-  I'^  ^i^'^t'  manner  the 
equations  (•><">),  (37j  form  a  ])air  Avliich  diti'ei'  only  in  the  sign  of  ii. 
Hence  if  //„  and  //_„  denote  pai'ticular  inti^grals  of  the  first  and  second 
res})e(ttively,  the  complt-tc  integrals  are  ri'S})ectively 

y  =  a //,,  +  ////_  ,_.,•-  - "     and     y  =  <<  11  _  „  +  />/A„:r  "  ; 

and  similai'ly  the  I'cspective  integrals  of  (38 j,  (39)  are 

y  =  <i  /„  +  // 1_  ,,/■ " "     a  n  d     y  =  ,iI_^^-\-  h  /„,-,■», 

where  /„  and  /_„  denote  jjarticular  integrals  of  these  two  equations.  It 
sh(juld  l)e  noted  that  these  forms  are  the  complete  solutions  only  when 
the  two  integrals  are  indtqiciidrnt.    Xote  that 

/,/.'•;)  =  .'■""  ^ "'  '„(  2  v;),      ( •„(./•)  =  (1  ,/•)"/„(  1  ..•-).        (40) 

As  it  has  1)een  seen  that  (\^-:=  {—\  fC _^^  when  n  is  integral,  it  follows 
that  in  this  case  the  aliovt-  forms  do  not  give  the  conq)lete  solution. 

A  particular  solution  of  (38)  may  readily  be  obtained  in  series  by  the 
method  of  undetermined  eoeiticients  (§  88j.    It  is 

1  ix)  =  y  r'X\  (I-  = -    -'*- ,  (41) 

as  is  derived  below.  It  sliould  be  noted  that  T_,^  formed  ly  clianging 
the  sign  of  n  is  meaningless  when  n  is  an  integer,  i'or  the  reason  that, 


250  DIFFERENTIAL  EQUATIONS 

from  a  certain  point  on,  the  coefficients  «;  have  zeros  in  the  denominator. 
The  determination  of  a  series  for  tlie  second  independent  solution  when 
n  is  integral  will  be  omitted.  The  solutions  of  (35),  (36)  corresponding 
to  /„(^')  are,  by  (40)  and  (41), 

»-V,(.T)=5;i^/,.(lA  (42') 

where  the  factor  n !  has  been  introduced  in  the  denominator  merely  to 
conform  to  usage.*  The  chief  cylinder  function  C'„(x)  is  Jn{x)  and  it 
always  carries  the  name  of  Bessel. 

To  derive  the  series  for  /„(x)  write 


1 
(1  +  n) 

X 


I„  =  tty  +     a^x  +         a._>.x2  +  •  •  •  +  ak-\x^~'^  +  •  •  • , 

7,^  =  a^  4-  2  a,,x  +       3  a^x^  ^ +  (A:  —  1)  a^ _ix*'  -  ^  4.  . . .  ^ 

/;;  =  2  rt.^    +  3  •  2  «,jX   +•••  +  (/£-  1)  (A;  -  2)  at  -ix^"-  »  + 


0  =  [«y  +  «i(u  +  1)]  +  X  [a^  +  a.^  {n  +  2)]  +  x^  [a,  +  a^}i  {n  +  3)] 
+  •  ■  •  +  x^-i[«A;-i  +  akk{n  +  A-)]  +  .  . . . 

Hence     a^  +  a-j()i  +  1)  =  0,     a^  +  032  (?i  +  2)  =  0,  •  •  • ,     a^  _i  +  aA-t  (n  +  A;)  =  0, 


a„  = 


^  )i  +  1 '  -       2  (ii  +  2)       2  !  (?i  +  1)  [n  +  2) '        ' 

^■ !  (u  +  1)  •  •  •  (n  +  i-) 

If  now  the  clioice  a^  =  1  is  made,  tlie  series  for  /„(/)  is  as  given  in  (41). 
The  famous  differential  eijuation  of  tlie  first  order 

^y'  —  uy  +  ^'//'"  =  ex",  (43) 

known  as  Eiccati's  equation,  may  be  integrated  in  terms  of  cylinder  functions. 
Note  that  if  n  =  0  or  c  =  0,  the  variables  are  separable  ;  and  if  h  =  0,  the  equation 
is  linear.  As  these  cases  are  immediately  integrable,  assume  ben  ^  0.  By  a  suitable 
change  of  variable,  the  equation  takes  the  form 

^^+  (^--)-ji-^'^V  =  0,         f  =  -;X'S         y  =  --l^-.  (430 

d^~       \        nl  «|  n-  ba^-q 

A  comparison  of  this  with  (39)  shows  that  the  solution  is 

a 

7,  =  A  /_  „  (-  hc^)  +  BI„{-  bc^)  ■  (-  bc^Y  , 

n  n 

which  in  terms  of  Bessel  functions  ,/  becomes,  by  (40), 

7;  =  i^  [.1.AJ2  V-7.^)  +  r>.T  ,,(2  V-  6cO]. 


*  If  n  is  not  integral,  both  n\  and  {a  +  i) !  must  be  replaced  (§  147)  by  r(«  +  1)  and 

r(/i  +  ^  +  i). 


ADDITIOXxVL   ORDINARY   TYPES  251 

The  value  of  y  may  be  found  by  substitution  and  use  of  (29). 

n       n 

„  ,/„      (2x2V-  hc/n)  -  AJ^    „(2x2  V-  hc/n) 

^=\-l^^^^ ^'^-n '  (44) 

J„(2x2  V-  hc/n)  +  A  J  „(2x-V-  hc/n) 

n  n 

where  A  denotes  the  one  arbitrary  constant  of  integration. 

It  is  noteworthy  tliat  the  cylinder  functions  are  sometimes  expressible  in  terms 
of  trigonometric  functions.    For  when  n  =  \  the  ecjuation  (35)  has  the  integrals 

y  =  A  sinx  +  7)  cosx     and     y  =  x^[ACi{x)  +  BC_  i(x)]. 

Hence  it  is  permissible  to  write  the  relations 

X  2  Ci  (x)  =  sin  X,        X  •;  C'_  i  (x)  =  cos  x,  (45) 

where  C  is  a  suital)ly  chosen  cylinder  function  of  order  \.  From  these  equations 
by  application  of  (29)  the  cylinder  functions  of  order  p  +  |,  where  p  is  any  integer, 
may  be  found. 

Now  if  Riccati's  equation  is  such  that  h  and  c  have  opposite  signs  and  a/n  is 
of  the  form  p  -\-  \i  the  integral  (44)  can  be  expressed  in  terms  of  trigonometric 
functions  by  using  the  values  of  the  functions  C  ^  j  just  found  in  place  of  the  J's. 
Moreover  if  h  and  c  have  the  same  sign,  the  trigonometric  solution  will  still  hold 
formally  and  may  be  converted  into  exponential  or  hyperbolic  form.  Thus  Riccati's 
equation  is  integrable  in  terms  of  the  elementary  functions  when  u/n  =  p  ■{■  h  wo 
matter  what  the  sign  of  he  is. 

EXERCISES 

1.  Prove  the  following  relations: 

{a)  4  C;  =  C„_2-2  C„  +  0«  +  2 ,         (^)  xC„  =  2{n+  1)  C„  +i  -  xC„  +  2, 

(7)  23C;;'  =  C„  _  3  -  3  C„ _i  +  3  C„  +,  -  C„  + , ,         generalize, 

(5)  xCn  =  2()i  +  1)  C„+i  -2{n  +  3)  C„  +  3  +  2  (ji  +  5)  (',,  +  5-  xC'„  +  6. 

2.  Study  the  functions  defined  by  the  pair  of  relations 

F„_i(x)  +  F„+i(x)  =  2^F„{x),         F„_i(x)  -  F„+i(x)  =  ^F„(x) 
ax  X 

especially  to  find  results  analogous  to  (30)-(35). 

3.  Use  Ex.  12,  p.  247,  to  obtain  (34)  and  tlie  corresponding  relation  in  Ex.  2. 

4.  Show  that  the  solution  of  (38)  is  y  =  AI„  f     ^  ^'\  +  BI„. 

n 

5.  Write  out  five  terms  in  the  expansions  of  I^,  I^,  I_i ,  J^,  J^. 

/2  1 

6.  Show  from  the  expansion  (42)  that  I  I  \  -Ji  (x)  =  ~  sin  x. 

\  X     i  X 

7.  From  (45),  (29)  obtain  the  following  : 

x^  Cs  (j)  = cos  X,  X  a  Cs  (x)  = 1 1  sm  x cos  x, 

2  X  2  \x"         /  X 

1^      ,  ,             .           cosx             iri      /  ^      3   .            /3        A 
X  2  C;_  5(x)  =  —  sni  X ,  X2  (7_  5  (x)  =  -  sm  x  +  ( 1  )  cos  x. 

2  X  2  X  \X-  / 


252  DIFFERENTIAL   EQUATIONS 

8.  Prove  by  integration  by  parts :    I  --  -  fZx  =  ~  +  6  ^  +  6  •  8      -^4-- 

J     x-^  X*  X*  J     £■' 

9.  Suppose  C„(j)  and  7\„(j")  so  chosen  that  A  =  1  in  (34).    Show  that 

V^A  Cn  (y )  +  BK„  (X)  +  L  r  K„  (X)  f  ^  dx  -  C„  (x)  J  ^^  cZxl 

is  the  integral  of  the  differential  equation  x"i/"  +  xy'  +  (x"  —  n-)y  =  Lx-^. 

10.  Note  that  the  solution  of  Hiccati's  e(ination  has  the  form 

^^^rf^^r.'     ^"'^  «l>«^v  that     '^l'  +  Pix)y.+  Q{x)y^=E(x) 
F{x)  +  A(,'{x)  dx 

will  be  the  form  of  the  e(iuation  wliich  lias  such  an  expression  for  its  integral. 

11.  Integrate  these  eiiuations  in  terms  of  cylinder  functions  and  reduce  the 
results  whenever  possible  by  means  of  Ex.  7  : 

(a)  xy'  -  '>y  +  y-  +  x^  =  0,         (/3)  xy'  -  3  ?/  +  y"  =  .>■"-. 

(7)  y"  +  ye-''  =  0,  (5)  x'^y"  +  nxy'  +  (h  +  rx-^'")y  =  0. 

12.  Identify  the  functions  of  Ex.  2  with  the  cylinder  functions  of  ix. 

13.  Let  (x2  -  1)  7^;  =  (n  +  1)  (P„  +1  -  xP„),         ^1  +1  =  ^K  +  '«  +  1)  ^«       (46) 

be  taken  as  defining  the  Legcndre  functions  P„{x)  of  order  n.    Trove 

(a)   (x2  -  1)  r;,  =  n  (xPn  -  r„  _i),        (^)   (2  n  +  1)  xP„  =  [n  +  1)  P,,  +i  +  '/  P„  _i , 
(7)   (2  n  +  1)  P„  =  P:  +1  -  P;_i,         (5)   (1  -  x--^)  P;;  -  2xP:  +  n{n  +  1)  /'„  =  0. 

A  A 

14.  Show  that  I'nQn  —  P„Q„  = and     P„Q„+i  -  Pn  +  iQn  = > 

X-  —  1  ?i  +  1 

where  P  and  Q  are  any  two  Legendrc  functions.    Express  the  general  solution  of 
the  differential  equation  of  Ex.  1.3  (5)  analogously  to  Ex.  4. 

15.  Let  u  =  X-  —  1  and  let  D  denote  differentiation  by  x.    Sliow 

J>"+iw"+i  =  7>" +!(»»")  =  uT)'>  +!«"  +  ■2{n  +  l)xTJ"u"  +  n  (n  +  1)  Jl"-h(", 
7;«+i„n+i  =J)nZ)un+i  =  ■)  (n  +  l)J)"{xu")  =  2  (h  +  l)xl)"u"  +  2  )i  (u  +  l)7>"'-i"". 

Hence  show  that  the  derivative  of  the  second  ecination  and  the  eliniinant  of  P"-i«'' 
between  the  two  ecpiations  give  two  e(iuations  winch  reduce  to  (Ki)  if 

„  1        d"      -,  ^^Vllen  ».  is  integral  these  are 

A,  .'•)  = (x2-])". 

•2"  ■  }i\  dx"  [Legendrc  H  ponjnoniKil!^. 

16.  Determine  the  solutions  of  Ex.  13  (S)  in  series  for  the  initial  conditions 

(a)  /'„(0)  r^z  1,    r;,fO)  :=  0,         {ji)  pjo)  =  0.    7';(0)  ==  i. 

17.  Take  /',,  --=;  1  and  I\  =  x.    S'low  tliat  tliese  are  solutions  of  (40)  and  compute 
P.,.  P.,.  /'.,  from  Iv\.  13  {;i}.    \i'  x  ~  cos  (9.  sliow 

P.,  =  ^cos2  0  +  I  P..  .r  Jcos3^+  gcos^.  P^  =  |J(.,,s4^+  |$cos2^  +  ^. 

18.  Write  Ex.  13  (5)  as  '    [  ( 1  -  x-)  7',',]  +  n  (n  +  1)  P„  =  0  an<l  show 

r  - '  r  *  1 '        fZ  ( 1  -  X-- )  7 ''  rf  ( 1  -  x2 )  P' "I 

[„,  („,  +  ])  _  „,(n  +  1)]  /       P„P,„dx  ^    /       I  P,„ -  Pn , ~   dx. 

J-i  J -I    1  dx  dx 


ADDITIONAL   DKDIXAKV   TYPES  253 

Integrate  by  pails,  assume  (he  fuiietions  ami  their  (lerivative.s  are  finite,  and  show 
I        l'„P,ndx  =  0,     it     n  ^  m. 

19.  By  suceessivc  int(\urati(iii  liy  parts  and  hy  reihiction  foi-niidas  show 
/-+!  ]  p+i  ,j„,f2  _iY,    ,]"i.i-~ —A]"  (—1)"    r+^ 

J -I  2-"()t  !)-J-i  (/,<:"  (k"  2«.)i!J-i 

/- -1 1     ,  2 

and  I       P'-dx  — ,  k  integral. 

J_i  2)1+1 

r  M  r  M      r?"(.i'-  —  IV' 

20.  Show  /       x'"PAf  =    I       •'•'"  =-0,         if  m<n. 

J-l  J-i  (/x" 

Determine  the  value  of  (he  integral  when  m  =  ?(.  Cainiot  the  results  of  Exs.  18,  19 
for  )ii  and  u  integral  be  obtained  sini])ly  from  tliese  I'esidts  ? 

./■'"  x*^  x'^ 

21.  Consiih'r   (38)   and   its   solution   Z,,  =  1  —  x  +  -^  ,-  —  — ^  ""I ^  —  .  .  •   when 

n  =  0.    Assume  a  solution  of  the  form  t/  —  l^^^•  +  w  so  that 

(I-m      (Jin  (U..(Iv       ^       ..        <l-v      dv 

X  -  -     +         +  ir  +  2  ./•     "  -  -  =  0,       if      .c    -  -  +  —  =  0, 
(/.(•-       dx  dx  dx  dx'~      dx 

is  the  eipiation  for  ;'.'  if  v  satisfies  the  e(piati(.)ii  xv"  -\-  r'  =  0.    Show 

„         ,  „      2  Bx      2  Rr~      2  Bx^ 

V  =z  A  +  ]}  lou'  X,         xir"  -f  w'  +  w  =  2  /> 1- +  •  •  • . 

2  !         2  !  ;3  !       :-5  !  4  ! 

By  assuming  id  =  a^x  -\-  ((.,x-  +  •  •  • ,  determine  the  «"s  and  henee  obtain 

and  (,1  +  7)  log.c)  7,1 -f  w  is  then  the  complete  solution  containing  two  constants. 
As  vl /|,  is  one  solution.  7ilog,/'  •  7,,  +  v  is  anotlier.  From  this  second  solution  for 
n  -  0,  the  st'cond  solution  f(U'  any  integral  value  of  ?(.  may  be  obtained  by  differ- 
entiation ;  the  worlc,  however,  is  long  and  the  result  is  somewhat  complicated. 


CHAPTER  X 

DIFFERENTIAL  EQUATIONS  IN  MORE  THAN  TWO  VARIABLES 

109.  Total  differential  equations.    An  equation  of  the  form 

P  {a;  ij,  z)  dx  +  Q  (r,  y,  z)  dy  +  R  (:r,  y,  z)  dz  =  0,  (1) 

involving  tlie  differentials  of  three  varial)les  is  called  a  fofal  differen- 
t'tdl  equation.  A  similar  equation  in  any  nurnVnu'  of  variables  would 
also  ])e  called  total;  hut  the  discussion  here  Avill  be  restricted  to  the 
case  of  three.  If  dehnite  values  be  assigned  to  x,  y,  z,  say  a,  h,  r,  the 
('(juation  becomes 

Adx  +  Bdy  +  Cdz  =  A  (x  -  a) -\-  B (y  -  h)  +  C (z  -  c)  =  0,       (2) 

where  x,  y,  z  are  supposed  to  l)e  restricted  to  values  near  a,  h,  c,  and 
represents  a  small  portion  of  a  plane  })assing  through  (a,  !>,  c).  From 
the  analogy  to  the  lineal  element  (§  85j,  such  a  ])ortion  of  a  plane  may 
be  called  a  plannr  element.  The  differential  ecpiation  tlierefore  repre- 
sents an  infinite  number  of  planar  elements,  one  passing  through  each 
point  of  space. 

Now  any  family  of  surfaces  F{x,  y,  z)  =  C  also  represents  an  iiifinity 
of  planar  elements,  namely,  the  portions  of  the  tangent  planes  at  every 
point  of  all  the  sui-faces  in  the  neighi)orhood  of  their  respective  points 
of  tangency.    In  fact 

dF  =  F;.dx  +  F'ydy  +  F^Iz  =  0  (3) 

is  an  equation  similar  to  (1).  If  the  planar  elements  represented  by 
(1)  and  (3)  are  to  be  the  same,  the  equations  cannot  differ  by  more 
than  a  factor  fJ^(x,  y,  z).    Hence 

f;  =  fip,      f;  =  /iQ,      f:  =  /^r. 

If  a  function  F(x,  //,  -')  =  f  can  be  found  which  satisfies  these  condi- 
tions, it  is  said  to  l)e  the  integral  of  (1),  and  the  factoi-  /j,  (.r.  y,  z)  by 
which  the  equations  (1)  and  (oj  differ  is  called  an  hitiyz-tifr/iy  picfur 
of  (1).    Compare   §  91. 

It  may  ha})])en  that  yu,  =  1  and  that  (l)  is  tlius  an  e.i'<ict  differential. 
In  this  case  the  conditions 


MORE  THAN  TWO  VARIABLES  255 

which  arise  from  F",j  =  l-"',!,,  F',^,  =  1-''"^,  F'^',.=  F^'^,  must  Ije  satisfied. 
Moreover  if  these  conditions  are  satisfied,  the  equation  (J)  will  be 
an  exact  equation  and  the  integral  is  given  by 

F{x,  y,  z)  =  f  P  (^,  y,  ^)  dx-\-  j     Q  (x^,  y,  z)  dy  +  \R  (.r^,  y,, .-)  dz  =  C, 

where  x^,  y^,  z^  may  be  chosen  so  as  to  render  the  integration  as  simple 
as  possible.  The  proof  of  this  is  so  similar  to  that  given  in  the  case  of 
two  variables  (§  92)  as  to  be  omitted.  In  many  cases  which  arise  in 
practice  the  equation,  though  not  exact,  may  be  made  so  by  an  obvious 
integrating  factor. 

As  an  example  take  zxdy  —  yzdx  +  xMz  =  0.    Here  the  conditions  (4)  are  not 
fuliilled  but  the  integrating  factor  l/x'-z  is  suggested.    Then 


xdy  —  ydx      dz  _     (y 


^>°-) 


X-  z 

is  at  once  perceived  to  be  an  exact  differential  and  the  integral  is  y/x  +  logz  =  C. 
It  appears  therefore  that  in  tliis  simple  case  neither  the  renewed  application  of  the 
conditions  (4)  nor  the  general  formula  for  the  integral  was  necessary.  It  often 
happens  that  both  the  integrating  factor  and  the  integral  can  be  recognized  at  once 
as  above. 

If  the  equation  does  not  suggest  an  integrating  factor,  the  question 
arises.  Is  there  any  integrating  factor  ?  In  the  case  of  two  variables 
(§  94)  there  always  was  an  integrating  factor.  In  the  case  of  three 
variables  there  may  be  none.    For 


cu.  cP  „  CLi  cQ 

■'          cy  cy  ex  ex 

dix  cQ  „  da  cR 

da  cR  „  ca  dP 

ex  ex  ~  cz  cz 


R, 

P, 

Q. 


If  these  equations  be  multiplied  l)y  A',  /',  (2  and  added  and  if  the  result 
be  simplified,  the  condition 

\cz        cyj  \cx       cz)  \cy       dx  J 

is  found  to  be  imposed  on  P,  Q,  R  if  there  is  to  be  an  integrating  fac- 
tor. This  is  called  the  condition  of  integrabUify.  For  it  may  be  shown 
conversely  that  if  the  condition  (5)  is  satisfied,  the  equation  may  be 
integrated. 

Suppose  an  attempt  to  integrate  (1)  be  made  as  follows  :  First  assume 
that  one  of  the  variables  is  constant  (naturall}',  that  one  Avhich  will 


256  DIFFERENTIAL   EQUATIONS 

make  the  lesultiiig  equation  simplest  to  integrate),  say  .'.■.  Tlien 
Pdx  +  Qdij  =  0.  Now  integrate  tliis  simplified  equation  Avitli  an  iiite- 
grating  factor  or  otherwise,  and  let  F{r,  ij,  .^■)  =  <^0v)  he  the  integral, 
when;  the  constant  (,'  is  taluai  as  a  function  ^  of  a.  Next  try  to  deter- 
mine <^  so  tliat  the  integral  F{.>',  ij,  '-)  =  ^  (■•)  '^v'ill  satisfy  (1).  To  do 
this,  dilferentiate ; 

F'jlx  +  F,////  +  Fjh  =  (/(f>. 

Compare  this  e(piation  Avitli  (1).    Then  the  e(;[uations'* 

i^';  =  xp,        /<;;  =  xq,        (/■':  -  xii)  >/.■:  =  ,icj> 

must  hold.  Tlu'  third  ecpiation  (F'^  ~-  XR)  th:  —  tl<^  may  he  integrated 
provided  the  coetH(dt',nt  S  ~--  y',  —  XI!  of  '/,-;  is  a  function  of  r:  and  </>, 
that  is,  of  ,'j  and  F  alone.  This  is  so  in  case  the  condition  (."))  holds.  It 
therefore  appears  that  the  integration  of  the  equation  (1)  for  whi(;h  (5) 
holds  reduces  to  the  succession  of  two  integrations  of  the  type  discussed 
in  Chap.  VIII. 

As  an  example  take  (2x-  +  2xi/  +  '2xz-  +  ])(lx,  +  dij  +  2zdz  =  0.   The  condition 

(2x--^  +  2xy  +  2XZ-'  +  1)0  +  1  (-  Axz)  +  22(2.r)  =  0 

of  intc'srability  is  satisfied.  Tlie  ^-reatcst  simplification  will  be  had  by  making  x 
constant.    Then  <bj  +  2  zdz  =  0  and  y  -{■  z~  =  (f>  (.c).    Compare 

dy  +  2  zdz  =  d(t>     and     (2  x-  +  2  xy  +  2  xz~  +  1)  dx  +  '///  +  2  zdz  -  0. 

Then  \  =  1,  -  (2  .)•-  +  2  ,/•//  +  2  xz-  +  ] )  dx  =  d.t,  ; 

or  —  (2  X-  +  1+2  X0)  dx  =  (Z^/.     or     dtp  +  2  x^l^dx  =r  -  (2 ./;-  +  ])  (/x. 

This  is  the  linear  ty^x;  witli  tlic  integral ing  factor  c''.   Then 

c'-{d<p  +  2 x^pdx)  =  -  c''(2 X-  +  \)dx      or      r ■  >/'  =  -  ( '''('- ••'"  +  1 ) '^■''  +  (^'■ 

Hence  //  +  ;•-  +  c^''  fc'-V-x-  +  l)dx  -  C'c--'-'  or  f'-'(//  +  z~)  +   Cc''\2x-  +  1)(/.i- ;..  6' 

is  the  solntion.  It  may  bo  noted  thai  c''  is  the  integrating  fa('t(n'  for  the  originid 
eqnalion  : 

t'--[(2x-  +  2x/y  +  2x~-^  +  ])dx  +  dy  +  2 zdz]  =  dlc'''{y  +  z-)  +  J<  -(2„--  +  ])dx  I  • 

To  complete  the  ]iroof  that  the  ('([nation  (1)  is  inlegrable  if  (."))  is  siitisfied.  ii  is 
necessaiy  to  show  that  wlien  the  cdiidilion  is  siilislied  the  cnelhcicnt  N  -.-.  /■','  —  X/,' 
is  a  fnnction  of  z  and  F  alone.  Let  it  be  i-egarded  as  a  fnnctiim  of  x,  F,  z  instead 
of  X,  y,  z.  It  is  necessary  to  ]irove  that  the  derivative  of  N  by  x  wlien  F  and  z  are 
constant  is  zero.    I>y  the  fornndas  for  change  of  vai'iable 

rx/,,,~      \(J''r,~      \f"/'7  (X  \(y'.,,z      \f"/'7j, c  '!/ 

•■■  Here  t'lie  factor  X  is  not  an  integrating  facturof  (li,  l)rit  only  of  tlie  reduced  e(]iiali(in 
/'(/.-•  +  (j</y  -  0. 


]\L()HE   THA^'    TWO  VAKIATU^ES  257 


But         F;  =  \P  and  Fj  =  \Q,  and  hence  Q  ryj      -  P  i'^}'\      =  Q  rpj 


/cS\  (    IcV       ^     \       c-F       c\I!       c\P      c\R 

Now  I )         =  —  I X/i  I  =: = 

\cx/„,  c      cx\cz  I       cZ(X        ex         cz         ex 

Hence  ^      =  X  (^  -  ^U  i>  ^  _ /.  ^, 

\cx/,,^~         \cz        ex]  cz  ex 

and  (^)      =x(^_^WQ^-i^^. 

\eij/.r,z         \ez       ey  I  cz  cy 

\exl„,,  \cy/x,z        L     \ez        ex  I  \ey        cz  J  A  \_     ex  cyA 

\cx/f,z        L     Vc::        cx/  \cy       cz  j  \ex       cy  I  \ 

Vex         ey  \ 

where  a  term  lias  been  added  in  tlie  first  bracket  and  subtracted  in  the  second. 
Now  as  X  is  an  integrating  factor  for  Vdx  +  (Idy,  it  follows  that  {\(i)\.  —  (XP)^,  ;  and 
only  the  first  lirackct  remains.  By  the  condition  of  iute^rability  this,  too,  vanishes 
and  hence  ^'  as  a  function  of  j,  F,  z  does  not  contain  x  but  is  a  function  of  F  and 
z  alone,  as  was  to  be  proved. 

110.  It  lias  bt'on  seen  that  if  tlie  equation  (1)  is  integrablo,  tliere  is 
an  integratiuL;-  i'aetor  and  the  condition  (5)  is  satisfied ;  also  that  con- 
versely' if  the  condition  is  satisfied  the  equation  may  be  integrated. 
GeonietriciiUy  this  tueans  that  the  infinity  of  ])lanaT  elements  defined 
oy  the  equtition  can  be  grouped  upon  a  family  of  surfaces  F{\v,  i/,  -S)  =  C 
to  which  they  are  tangent.  If  the  condition  of  integnil)ility  is  not  satis- 
fied, the  phuiar  elements  cannot  be  thus  grouped  into  surfaces.  Xever- 
theless  if  a  surface  G(,r,  //,  .■.)  =  0  l)e  given,  the  planar  element  of  (1) 
which  passes  through  any  point  (.r^,  //^,  ,-.\,)  of  the  surface  will  cut  the 
surface  G  =  0  in  a  certain  lineal  elen'ient  of  the  surface.  Thus  upon  the 
surface  (/  (.'•,  //,  z)  =  0  there  will  l)e  an  infinity  of  lineal  elements,  one 
through  each  point,  which  satisfy  the  given  equation  (1).  And  these 
elements  niay  lie  grouped  into  curves  lying  upon  the  surface.  If  the 
equation  (1)  is  integrable,  these  curves  will  of  course  be  the  intersections 
of  the  given  surface  6^  =  0  with  the  surfaces  F  =  C  defined  by  the 
integral  of  (1). 

The  method  of  obtaining  the  curves  upon  6' (,'■,  //,  ,v)  =  0  which  are 
the  integrals  of  (1),  in  case  (5)  does  not  possess  an  integral  of  the  form 
F(.r,  I/,  ,-')  =  C,  is  as  follows.    Consider  the  two  equtitions 

]>J,r  +  QJif  +  /.V/,v  =  0,  r/>/,r  +  ^'^/y  -J-  G^dz  =  0, 

of  which  tlie  first  is  the  gi\'en  diiferential  (>(piation  and  the  second  is 
the  ditfereiitial  e(_[uatioii  of  the  gi\-eu  surface.    Froiu  these  etptatioiis 


258  DIFFERENTIAL  EQUATIONS 

one  of  the  differentials,  say  dz,  may  be  eliminated,  and  the  correspond- 
ing variable  z  may  also  be  eliminated  by  substituting  its  value  obtained 
by  solving  G  (x,  ?/,  .^')  =  0.  Thus  there  is  obtained  a  differential  equa- 
tion Mdx  -\-  N(h/  =  0  connecting  the  other  two  variables  x  and  y.  The 
integral  of  this,  F{x,  y)  =  C,  consists  of  a  family  of  cylinders  which  cut 
the  given  surface  C  =  0  in  the  curves  which  satisfy  (1). 

Consider  the  equation  ydx  +  xdy  —  {x  +  y  +  z)  dz  =  0.  Tliis  does  not  satisfy  the 
condition  (5)  and  lience  is  not  completely  integrable  ;  but  a  set  of  integral  curves 
may  be  found  on  any  assigned  surface.    If  the  surface  be  the  plane  z  =  x  +  y,  then 

ydx  +  xdy  —  {x  +  y  +  z)dz  =  0     and     dz  =  dx  +  dy 

give  (x  +  z)dx  +  {y  +  z)  dy  =  0     or     {2x  +  y)dx  +  {2y  +  x)dy  =  0 

by  eliminating  dz  and  z.    The  resulting  equation  is  exact.    Hence 

x^  +  xy  +  y"^  =  C     and     z  =  x  +  y 

give  the  curves  which  satisfy  the  equation  and  lie  in  the  plane. 

If  the  equation  (1)  were  integrable,  the  integral  curves  may  be  used  to  obtain 
the  integral  surfaces  and  thus  to  accomplish  the  complete  integration  of  the  equa- 
tion by  Mayefs  method.  For  suppose  that  F{x,  y,  z)  =  C  were  the  integral  surfaces 
and  that  F{x,  y,  z)  —  F{0,  0,  Zq)  were  that  particular  surface  cutting  the  z-axis  at  z^. 
The  family  of  planes  y  —  \x  through  the  z-axis  would  cut  the  surface  in  a  series 
of  curves  which  would  be  integral  curves,  and  the  surface  could  be  regarded  as 
generated  by  these  curves  as  the  plane  turned  about  the  axis.  To  reverse  these 
considerations  let  y  =  'kx  and  dy  =  'Kdx  ;  by  these  relations  eliminate  dy  and  y  from 
(1)  and  thus  obtain  the  differential  equation  Mdx  +  Ndz  =  0  of  the  intersections 
of  the  planes  with  the  solutions  of  (1).  Integrate  the  equation  as/(x,  z,\)  =  C  and 
determine  the  constant  so  that/(x,  z,  X)  =/(0,  Zq,  X).  For  any  value  of  X  this  gives 
the  intersection  of  F{x,  y,  z)  =  F{0,  0,  z„)  with  y  =  Xx.  Now  if  X  be  eliminated  by 
the  relation  X  =  y/x,  the  result  will  be  the  surface 

/|x,  z, -|  =/|0,  Zq, -|,     equivalent  to     i^(x,  ?/,  z)  =  i''(0,  0,  Zq), 

which  is  the  integral  of  (1)  and  passes  through  (0,  0,  Zg).  As  z^  is  arbitrary,  the 
solution  contains  an  arbitrary  constant  and  is  the  general  solution. 

It  is  clear  that  instead  of  using  planes  through  the  z-axis,  planes  through  either 
of  the  other  axes  might  have  been  used,  or  indeed  planes  or  cylinders  through  any 
line  parallel  to  any  of  the  axes.  Such  modifications  are  frequently  necessary  owing 
to  the  fact  that  the  substitution  /(O,  z^,  X)  introduces  a  division  by  0  or  a  log  0  or 
some  other  impossibility.    For  instance  consider 

y^dx  +  zdy  —  ydz  =  0,         y  =  Xx,         dy  —  \dx,         X^x^dx  -|-  Xzdx  —  \xdz  =  0. 

Then  Xdx  H ^  =  0,     and     Xx =/(x,  z,  X). 

x'^  X 

But  here  /(O,  z,,,  X)  is  impossible  and  the  solution  is  illusory.  If  the  planes  {y—l)  =  \x 
passing  through  a  line  parallel  to  the  z-axis  and  containing  the  point  (0,  1,  0)  had 
been  iised,  the  r(;sult  would  bo 

dy  =  "Kdx,         (1  4-  \x)'-dx  +  \zdx  —  {I  +  \x)dz  =^  0, 


MORE  THAN  TWO  VARIABLES  259 

or  dx -\ ^^ —  =  0,     and     x =/(x,  z,  X). 

(1  +  \j)2  '  1  +  Xx      ^  '    '    ' 

Hence  x =—  z,-,     or    x =—  Zn  =  C, 

1  +  Xx  "^  2/  '         ' 

is  the  solution.  Tlie  same  result  could  have  been  obtained  with  x  =  X^  or  y  =  \{x  —  a). 
In  the  latter  case,  however,  care  should  be  taken  to  use/(x,  z,  X)  =/(«,  z^,  X). 

EXERCISES 

1.  Test  these  equations  for  exactness  ;  if  exact,  integrate  ;  if  not  exact,  find  an 
integrating  factor  by  inspection  and  integrate  : 

(a)  (y  +  z)  dx  +  (z  +  x)(Zj/  +  (x  +  y)  dz  =  0,        (/3)  yhU  +  zdij  -  ydz  =  0, 

(7)  xdx  +  2/dy  —  Va-  —  x^  —  y'hlz  =  0,  (5)  2  z  (t/x  —  fZ^/)  +  (x  —  y)  dz  =  0, 

(e )  {•>x+  y-+  2 xz) dx  +  2 xydy  +  x-dz  =  0,      (f )  z;/fZx  =  zxdy  +  ?/"^(Z2, 

(7,)  x(y  -  1)  (z  -  l)rfx  +  2/(2  -  1)  (X  -  ^dy  +  z(x  -  1)  (i/  -  l)(iz  =  0. 

2.  Apply  the  test  of  integrability  and  integrate  these: 

(a)  (x^  -  ?/2  —  z2)  cZx  +  2  xydy  +  2  xzdz  =  0, 

(^)  (X  +  2/^  +  z-  +  1)  (Zx  +  2  2/tZ2/  +  2  z(Zz  =  0, 

(7)  {y  +  c)'^-^  +  ^(Zy  -{y  +  a)dz, 

(5)  (1  —  X-  -  2  2/2z)  fZz  =  2  xztZx  +  2  yz-tZ;/, 

( e )  x-dx-  +  2/"tZ2/-  —  z-(Zz"^  +  2  xydxdy  =  0, 

( f )  2  (xcZx  +  ydy  +  zdzf  =  (z^  —  x-  -  y^)  {xdx  +  ydy  +  zdz)  dz. 

3.  If  the  equation  is  homogeneous,  the  substitution  x  =  wz,  y  =  I'z,  frequently 
shortens  the  work.  Show  that  if  the  given  equation  satisfies  the  condition  of  inte- 
grability, the  new  ecjuation  will  satisfy  the  corresponding  condition  in  the  new 
variables  and  may  be  rendered  exact  by  an  obvious  integrating  factor.   Integrate  : 

{a)  (y-  +  yz)  dx  +  (xz  +  z")  dy  +  (2/2  -  xy)  dz  =  0, 

{(i)  (x-2/  —  2/^  —  y'^z)  dx  +  {xy-  -  x^z  —  x^)  dy  +  {xy-  +  x-y)  dz  =  0, 

(7)  (2/-  +  yz  +  Z-)  dx  +  (x^  +  xz  +  Z-)  dy  +  (x-'  +  xy  +  2/-)  dz  =  0. 

4.  Show  that  (5)  does  not  hold  ;  integrate  subject  to  the  relation  imposed  : 
{a)  ydx  +  xdy  —  (x  +  y  +  z)  dz  =  0,         x  +  y  +  z  =  k     or     y  =  kx, 

(/3)  c  {xdy  +  ydy)  +  Vl  —  a'^x'^  —  b'^y'^dz  =  0,         u-x'^  +  h-y-  +  c~z-  —  1, 
(7)  dz  —  aydx  +  bdy,         y  —  kx     or     x-  +  2/'  +  z-  —  \     or     y  =/(x). 

5.  Show  that  if  an  equation  is  integrable,  it  remains  integrable  after  any  change 
of  variables  from  x,  2/,  z  to  m,  r,  w. 

6.  Apply  Mayer's  method  to  sundry  of  Exs.  2  and  3. 

7.  Find  the  conditions  of  exactness  for  aii  ecjuation  in  four  variables  and  write 
the  fornu;la  for  the  integration.    Integrate  with  or  witiiout  a  factor  : 

{a)  (2x  +  y-  +  2xz)(Zx  +  2  xydy  +  x-dz  +  du  =  0, 

{j3)  yzudx  +  xzudy  +  xyudz  +  xyzdu  =0, 

(7)   {y  +  z+  u) dx  +  {x  +  z+  u) dy  +  (X  +  y  +  u) dz  +  {x  +  y  +  z) du  =  0, 

(5)  u{y  +  z)  dx  +  u{y  +  z  +  1)  dy  +  udz  -  {y  +  z)du  =  0. 

8.  If  an  equation  in  four  variables  is  integrable,  it  nmst  be  so  when  anj'  one  of 
the  variables  is  held  constant.  Hence  the  four  conditions  of  integrability  obtained 
by  writing  (5)  for  each  set  of  thi'ee  coefficients  uuist  hold.    Show  that  the  conditions 


260  DIFFEKEXTIAL   EQUATIONS 

are  .satisfied  in  the  following  cases.  Find  the  integi-als  by  a  generalization  of  the 
method  in  the  text  by  letting  one  variable  be  con.stant  and  integrating  the  three 
remaining  terms  and  determining  the  con.stant  of  integration  as  a  function  of  the 
fourth  in  such  a  way  as  to  satisfy  the  equations. 

(a)  z{y  +  z)  dx  +  z(u-  /)  dy  +  y  (x  -  u)  dz  +  y  {y  +  2)  du  =  0, 

(/i)   uyzdx  +  UZ.C  lug  jtdy  +  uxy  log  xdz  —  xdii  =  0. 

9.  Try  to  extend  the  method  of  Mayer  to  such  as  the  above  in  Ex.  8. 

10.  If  G'(r,  y,  z)  —  a  and  II {x.  y.  z)  —  h  are  two  families  of  .surfaces  defining  a 
family  of  curves  as  their  iiiter.sections,  show  that  the  equation 

(f;;/f;  _  aui;)dx  +  (a'jr.  -  G'ji:)dy  +  (rr//;  -  G'^ir:)dz  =  o 

is  the  eijuation  of  the  planar  elements  perpendicular  to  the  curves  at  every  point 
of  the  curves.  Find  tlie  eruditions  on  G  and  //  that  there  shall  V)e  a  family  of  sur- 
faces which  cut  all  these  curves  orthogonally.  Determine  whether  the  curves  below 
have  orthogonal  trajectories  (surfaces)  ;  and  if  they  have,  find  the  surfaces  : 

(a)  y  =  X  +  (I,  z  :=  X  +  h,  (P)'  V  -■  "■''  +  1.2=  hx, 

(7)  •*■-  +  y-  =  "-•  ^  =  '',  (5)  -ey  =  a.  xz  =  h. 

{ e)  x~  +  y-  +  z-  =  a-,  xy  =  h,  (j-)  /-  +  -1  y-  +  32-  =  <i,  xy  +  2  =  '>, 

(7?)  log  xy  =  «2,  X  +y  -\-  z  =  h,  (0)  y  = -2  ,tx  +  (/-,  z  =  •>  hx  +  //-'. 

11.  Extend  the  work  of  proposition  3,  §  04.  and  Ex.  11,  p.  234.  to  find  the  normal 
derivative  of  the  .solution  of  eijuation  (1 )  and  to  show  that  the  singular  solution  may 
be  looked  for  among  the  factors  of  /wi  =  0. 

12.  If  F  =  /-"i  +  Qj  +  /ik  be  formed,  show  that  (1)  becomes  F.(Zr  =  0.  .Show 
that  the  condition  of  exactness  is  VxF  =  0  by  expanding  VxF  as  the  formal  vector 
product  of  the  operator  V  and  the  vector  F  (see  vj  78).  Show  further  that  the  condi- 
tion of  integrability  is  F.(VxF)  =  0  by  similar  formal  expansion. 

13.  In  Ex.  10  consider  Vr  and  V//.  Sliow  these  vectors  are  normal  to  the  sur- 
faces G  =  a.  II  —  h.  and  hence  infer  that  (V^V)x(V//)  is  the  direction  of  the  inter- 
section. Finally  explain  why  (Zr.(V6'xV//)  =  0  is  the  differential  e(iuation  of  the 
orthogonal  family  if  there  be  such  a  family.  Show  that  this  vector  form  of  the  family 
reduces  to  the  form  above  given. 

111.  Systems  of  simultaneous  equations.    The  two  equations 

(Lr  ■  (l.f 

in  the  two  dependent  vuriabh-s  //  :uul  z  and  the  inde])en(h'nt  vaiialile  ,/• 
constitute  a  set  of  siniultaiu'ous  (Miimtions  of  the  lirst  orcUn'.  It  is  nnn'e 
customary  to  write  tlu'se  (^(luations  in  the  form 

which  is  synunetrie  in  tlie  differentials  and  wliere  A':  Y:Z  =  A  '■/'■',/■ 
At  any  assigned  ])oint  ./;.  //  .  :y  of  space  tlie  ratios  d.f  :  il i/ :  ilr:  of  the 
differentials  tire  determined  l>y  sulistitution  in  {~ ).    lleiu'c  the  eijutitions 


MOEE  THAX  TWO  VARIABLES  261 

fix  a  definite  direction  at  each  })oint  of  space,  that  is,  they  determine  a 
lineal  element  through  each  point.  The  problem  of  integration  is  to 
combine  these  lineal  elements  into  a  family  of  curves  /'(.'•,  //,  '-')  =  C\, 
('(■'■,  //,  r:)  =  t\„  de})cnding  on  two  parameters  C\  and  ('.,,  one  curve  ])ass- 
ing  through  each  })oint  of  S})ace  and  having  at  that  })oint  the  direction 
determined  by  the  e(piations. 

For  the  formal  integration  there  are  several  allied  methods  of  pro- 
cedure.   In  the  first  place  it  may  ha])pen  that  two  of 
(l.r       (Jji  (J  1 1       iJz  dx       (1r: 

y~y'       T^z'       T~z^ 

are  of  such  a  form  as  to  contain  only  the  variables  whose  differentials 
enter.  In  this  case  these  two  may  be  integrated  and  the  two  solutions 
taken  togethei-  give  the  family  of  curves.  Or  it  may  ha}>])en  that  one 
and  only  one  of  these  equations  can  l)e  integrated.  Let  it  be  the  first 
and  suppose  tliat  F{.i\  //)  =  C\  is  the  integral.  l)y  means  of  this  inte- 
gral the  varial)le  ./■  may  be  eliminated  fi'om  the  second  of  the  equations 
or  the  varial)le  //  from  the  third.  In  the  respective  cases  there  arises 
an  ecjuation  which  may  be  intcgi'atcd  in  the  form  (id/,  .'.,  C^^C,,  or 
G{.r,  •:,  F )  —  (\,.  and  tliis  result  taken  with  /''(■'';  //)  =  '"j  ^vill  determine 
the  family  of  curves. 

.rtJx       yd)/      (Iz 
Consider  the  example  =  '    '    =  —   Here  tlie  two  eijuatioiis 

yz         sz        y 

xdx      ydy  ,     xdx 

—  '  -'-     and     -      =  dz 

y  •'•'  z 

are  integrable  with  tlie  results  x^  —  y^  =  (\.  x-  —  z-  =  C.,,  and  these  two  inten-rals 

constitute  the  solutiim.    Tlie  solution  niiulit,  of  eourse,  appear  in  very  different 

form  ;  for  there  are  an  indetinite  nundier  of  ])airs  of  t'quations  F(x,  ?/,  z.  (\)  =  0, 

G  {x,  y.  z.  ('.,)  =  0  which  will  intei'sect  in  the  curves  of  intersection  of  x^  —  y^  =  (.', , 

and  /-  —  2-  =  r., .    In  fact  (//•"  +  (\)-  ^  {z-  -\-  C.,f  is  clearly  a  solution  and  could 

replace  eitlier  of  those  found  above. 

Consuler  the  example —    =      '     = Here 

.e-  —  //-  —  z-      2  xy      'J,  xz 

dii      dz  .  ,     ,      . 

—    —  -     ,     wuh  the  uitenTal     y  =  t  .z, 

is  the  only  equation  the  integral  of  which  can  be  obtained  directly.  If  y  be  elinn- 
nated  by  means  of  tliis  lirst  iiitciiral.  there  results  the  e(]uation 

'^] ='^     or     i>r-r?,r  +  [(rf+r)2--.r-]./-  =  0. 

./•--(rf+i)2-     -^sz 

This  is  honiogeiieous  and  may  be  intei:raled  witli  a  factor  to  uive 

.'•-  -VC"]^  1)  :-  -=  <'.z     or     x^  J^  y-i  j^  z-  =  C.z. 
Hence  //  —  (\z.         x~  +  y-  +  ,~-  =  ('.-,z 

is  the  sohaiiin.  and  represents  a  certain  fauulj"  of  circles. 


262  DIFFERENTIAL  EQUATIONS 

Another  method  of  attack  is  to  use  composition  and  division. 

dx  _  (hi  _  dz  __  \dx  +  /Ac///  -|-  vdz 

Y  ~  T  ~  "Z  ~    XX  +  fxY  +  vZ  ^^ 

Here  X,  /x,  v  may  be  chosen  as  any  functions  of  (,r,  y,  ?.-).  It  may  be 
possible  so  to  choose  them  that  the  last  expression,  taken  with  one  of 
the  first  three,  gives  an  equation  Avhicli  may  be  integrated.  AVith  this 
first  integral  a  second  may  be  obtained  as  before.  Or  it  may  be  that 
two  different  choices  of  A,  ix,  v  can  be  made  so  as  to  give  the  two  desired 
integrals.  Or  it  may  be  possible  so  to  select  two  sets  of  multipliers  that 
the  equation  obtained  by  setting  the  two  expressions  equal  may  be 
solved  for  a  first  integral.  Or  it  may  be  possible  to  choose  A,  /u.,  v  so 
that  the  denominator  AA'  -{-  /xY  -\-  vZ  =  0.  and  so  that  the  numerator 
(which  must  vanish  if  the  denominator  docs)  shall  give  an  equation 

Xdx  +  fjidf/  +  vdz  =  0  (9) 

which  satisfies  the  condition  (5)  of  integrability  and  may  be  integrated 
by  the  methods  of  §  109. 

Cunsider  the  equations = = Here  take  X,  fj.,  v 

^'  +  y-+  yz      X-  +  y-  -  xz      {x-{-  y)z 

as  1,   —  1,  —  1 ;  then   \X  +  fiY  +  vZ  =  0  and  dx  —  dy  —  dz  =  0  is  integrable  as 

X  —  y  —  z  =  C\.  This  may  be  used  to  obtain  another  integral.    But  another  choice 

of  X,  fi,  V  as  X,  y,  0,  combined  witli  the  hist  expression,  gives 

xdx  +  ydx  dz  ,      .  ,        o^      ,       o      ^ 

"  or     hjg  {x-  +  y-)  =  log  2-  +  Co . 


(x2  +  y^)  (X  +  y)       {X  +  y) : 
Hence  x  —  y  —  z  =  (\     and     ./■-  +  //-  =  C.^z^ 

will  serve  as  solutions.    This  is  shorter  than  the  method  of  elimination. 

It  will  be  noted  that  these  e(]uations  just  solved  are  homogeneous.    The  substi- 
tution X  =  iiz.  y  =  vz  might  be  tried.    Then 

udz  +  zdu,  _    vdz  +  zdv    _     dz     _         zdii         _         zdv 
u'^  +  v^  +  V      u-  +  V-  —  u      u  +  V      V-  —  uv  4-  V      u-  —  uv  —  u 
du  dv  dz 


v'^  —  uv  +  V      H-  —  uc  —  u       z 

Now  the  first  equations  do  not  contain  z  and  may  be  solved.    This  always  happens 
in  the  liomogeneous  case  and  may  be  employed  if  no  shorter  method  siigu-csts  itself. 

It  need  hardly  be  mentioned  that  all  these  methods  apply  equally  to 
the  case  Avhere  there  are  more  than  tlirec  ('(piations.  The  geometric^ 
})ictiire,  however,  fails.  altli(MiL;li  tlic  geometric  language  may  btM'ontin- 
ued  if  one  wishes  to  deal  witli  liiglicr  dimensions  than  three.  In  some 
cases  the  introduction  of  a  fourth  \-arial)le,  as 

(10) 


dx       <h,       d:: 

df 

dt 

: — — :   -     -   — 

~   

or 

— -    

A        Y       Z 

1 

t 

MORE  THAN   TWO  VAKIABLES  263 

is  useful  in  solving  a  set  of  equations  which  originally  contained  only 
three  variables.  This  is  }>articularly  true  Avhen  A',  Y.  Z  are  linear  with 
constant  coefficients,  in  wliich  case  the  methods  of  §  98  may  be  applied 
with  f  as  independent  varialjle. 

112.   Simultaneous  differential  equations  of  higher  order,  as 

(P.I-       ,.  /  <Jx    d>/\         (I't/       ,,  /  'Lr    (hi 


(It-    -^y '^'  dt  iUj      df       V       '/^  '^^ 

d-r         /dd>\-  /  (Ir    (/<b  \  Id/.,  dcb\  I  dr    dd, 

It'  -  '•(;l)  -  "  ('■■  *'  r,'i,)'     rJt  ('-  Tt)  =  *  ('■'  *'  ;77  ■  i 

especially  those  of  the  second  order  like  these,  are  of  constant  occur- 
rence in  mechanics  ;  for  the  acceleration  requires  second  derivatives 
with  respect  to  the  time  for  its  expression,  and  the  forces  are  expressed 
in  terms  of  the  coordinates  and  velocities.  The  complete  integration  of 
such  equations  requires  the  expression  of  the  de})endent  variables  as 
functions  of  the  indei)endent  variable,  generally  the  time,  with  a  num- 
ber of  constants  of  integration  e(jual  to  the  sum  of  the  orders  of  the 
equations.  Frequently  even  when  the  complete  integrals  cannot  be 
found,  it  is  })0ssible  to  carry  out  some  integrations  and  re})lace  the 
given  system  of  e(|uations  by  fewer  ecpiations  or  equations  of  lower 
order  containing  some  constants  of  integration. 

Xo  special  or  general  rules  Avill  be  laid  down  for  the  integration  of 
systems  of  higher  order.  In  each  case  some  particular  comljinations  of 
the  equations  may  suggest  themselves  which  will  enable  an  integration 
to  be  performed.*  In  ])roblems  in  mechanics  the  princi})les  of  energy, 
momentum,  and  moment  of  momentum  frequently  suggest  combinations 
leading  to  integrations.    Thus  if 

.■"  =  A,  y"=Y,  z"  =  Z, 

where  accents  denote  differentiation  with  resjject  to  the  time,  be  multi- 
plied by  </.'■,  <///,  dr:  and  added,  the  result 

X^d.r  +  ;/"d,j  -f  -"dz  =  Xdr  +  Vd,/  +  Zdz  (11) 

contains  an  exact  differential  on  the  left ;  then  if  the  expression  on  the 
right  is  an  exact  tlifferential,  the  integration 

h  (•'■"  +  .'/"'  +  ^'-)  =   f^Yd,'  +  Yd;/  +  Zdr:  +  C  (IV) 

*  Ir  is  possihlf  t((  ililTereiitiatc  the  .:,'ivcn  equations  repeateiUy  ami  eliniiiiate  all  the 
ih'iifiKlriit  \'ai-ial)li's  except  (iiie.  The  resulting  differential  e(piatiou.  sa>"  in  .!•  and  /,  may 
then  1)1-  treated  liy  the  methods  (jf  previous  chapters ;  but  this  is  rai'ely  suceessful  except 
when  the  equation  is  linear. 


yx 


2G4  DIFFEKENTIAL   EQUATIONS 

Ciiu  be  performed.  This  is  tJie  prlnrlpJc  of  ''ni'iu/y  m  its  siiui)lest  form 
If  two  of  the  equations  are  multi})lied  by  the  chief  variable  of  the  other 
and  subtracted,  the  result  is 

y.r"  -  .ry"  =  !/X  ^  xY  (12) 

and  the  expression  on  the  left  is  again  an  exact  differential ;  if  the 
right-hand  side  reduces  to  a  constant  or  a  function  of  f,  then 

is  an  integral  of  the  equations.  This  is  tlw  principle  of  mnmenf  of 
inoiaentiiiii.    If  the  e(|uati(jns  can  be  nuilti})lied  by  constants  as 

U"  +  /////"  +  n::"  =  IX  +  u,  Y  +  nZ,  (13) 

so  that  the  expression  on  the  right  reduces  to  a  function  of  t,  an  inte- 
gration may  be  performed.  This  is  tlin  prlnc'qjJe  of  inoiiientuiH.  These 
three  are  the  most  commonly  usaljle  devices. 

As  an  example  :  Let  a  particle  iihinc  in  a  plane  snljject  to  forces  attracting  it 
toward  tlie  axes  by  an  ainonnt  proportional  to  tin'  mass  ami  to  the  distance  from 
the  axes;  discuss  the  motion.    Here  the  ecpiations  of  motion  are  merely 

d-x  ,  d-ii  ,  d'-x  ,  d'-ii 

m  —  =  —  kmx,         m  — ^  =  —  kmij     or     —  =  —  kx,         — —  =  —  ky. 
dV^  df^  df^  dV^ 

Then      dx'^^^di/^^=-k{xdx  +  iidy)     and     i'^^^-\\{'^''\~  =  -k{x-+if)  +  C. 
dt-   '        dt^  \dtj        \dll 

Also  ?/ .'■  -   ■-  =  0     and     ?/ x  —  =  C  . 

df-  dC-  dt  dt 

In  this  case  the  two  principles  of  energy  and  moment  of  momentum  irive  two 
integrals  and  the  equations  are  I'educeil  to  two  (if  the  hi'st  order.  But  as  it  happens, 
the  original  e(iuations  couhl  be  integrated  directh'  as 

—  dx  =  —  kxdx.  (  =  -  kx-  +  (  -.  — =-^=  =  dt 

dV^  '  \'^^^  Vc^-kx^ 

dt-  \dlj  VK--ky^ 

Tlie  constants  ("-  and  7v"- of  integi'ation  have  been  written  as  sijuares  because  they 
are  necessarily  positive.    The  eomplctc  integration  gi\es 

Vkx  =  ('sin  (Vkl  +  (\),        \ki/  =  Jv'sin  (\kt  +  K.X 

As  another  example  :  A  ])artirh'.  attracti'il  toward  a  point  b_v  a  force  eijual  to 
r/ui-  +  //-/r"  pel-  uint  mass,  wiifre  //(  is  the  mass  and  A  is  the  (loul.)le  areal  Vfjocity 
and  r  is  the  distance  fiMm  the  ixjint.  is  ]ii-ojected  ])ei-pendicuhu'l y  to  tlie  radius  xcc- 
tor  at  the  distance  \  m/i  :  discuss  the  UKJtioii.  In  imhu'  c'lnirdiiiatcs  tlie  eiiuati.uis 
of  mot  inn  arc 

j '/■-/■  /d'l>\-\       ,,  itir       mil-  111  d  I  .,d(j>\ 


df- 


'  '  -//  '    I        '  ~~  ui-  ~    r   '  /•  dt\     dt  ' 


^l>  =  0. 


MOEE  THAN  TWO  VAKIA15LES  205 

The  second  integrates  directly  as  r-d<p/dt  =  h  where  tlie  constant  of  integration  h 
is  twice  the  areal  velocity.    Now  substitnte  in  the  lirst  to  eliminate  (p. 

d-r      li-  y        Ifi  d-r  r  /dr\~  r- 

— = ; J     or     -—  = or      -  -     = +  C. 

dt-       r"  ni-      r^  dt-  iii-  \dt/  m- 

Now  as  the  partick'  is  projected  perpendicularly  to  the  radius,  dr/dt  =  0  at  the 
start  when  r  =Vi>i/i.    Hence  the  constant  C  is  ]t/m.    Then 

dr  ,,  -     r''d(p       ,,       .  V)nhdr 

-  dt     anil     =  tie    give =  d^. 


Ii_4  '  .-  ,1 

\  ;/i      m-  \         h  m 


Hence  V//(A  \/   ,  —  =  (p  +  C     or =-'-' — ^. 

V  /■-       h  r-      Inn  mli 

Now  if  it  be  assumed  tliat  <^  =  0  at  the  start  \\  hen  r  =  xmli,  we  find  (7  =  0. 

o          ?"''■  .      , 

Hence  r-^  ^ is  the  orbit 

1  +  ^" 
To  find  the  relatit)n  between  0  and  the  time, 

T-dd>=]idt     or —  dt     nr     i  =  j?i,  tan-^rf), 

!  +  </>- 

if  the  time  be  taken  as  t  =  0  Vviien  4>  —  Q.  Thus  the  orbit  is  found,  the  expression 
of  4>  as  a  function  of  the  time  is  found,  and  the  expressidii  of  r  as  a  function  of  the 
time  is  obtainable.  'J'he  problem  is  completely  solved.  It  will  be  noted  th.at  the 
constants  of  integration  ha\e  l)een  determiiu'd  after  each  integration  by  the  initial 
conditions.  'I'his  simplifies  the  subse(iuent  integrations  which  might  in  fact  be 
impossible  in  tt  rms  of  elementary  functions  \Nitiiout  this  simplification. 


EXERCISES 


1.  Integrate  these  equations: 


dx      di/       dz  dx      dy  dz 

ijz      xz      xij  y-       X-      x-y-z- 

dx      dy      dz  ,.,    dx      dy  dz 

(7)    —  r=  -      =  —  ,  (5)    —  =  —  =  — —  , 

XZ      yz      xy  yz      xz      x  +  V 

dx  _  (///  _     dz  dx    _       dy       __        dz 

^''   ~  y    "   X  ~f+^-'  ^^'   -T"   a7T4  5  72^+T2' 

dx  dy  dz 


2.  Integrate  ihe  ec|uations  :  («) 


^,"i  —  c/y      ex  —  HZ      ay  —  bx 


dx      __    dy   _       dz  dx     _^     dy     __     <lz 

x-  +  y-      2  xy      xz  +  yz  y  +  zx  +  zx  +  y 

dx         _         dy         __         dz  dx        _       dy                 dz 

y-x-2x*       -ly^-x'hi      z{x-^  -  y^)  ■''{y-z)       y(z-x}       z(x-y) 

.          dx                    di/                    dz  ,  .          dx                 —  di/                 dz 


x:{y--z-^]      y{z--x-^)      z(x--y~)  x(y-^-z-)      y{z^  +  x^)      z{x- ^  y-) 

dx           dy           dz          ,  ,        (/•''               dy                  dz             ,, 

=  — ^-^  =: =  dt,  (0 = = '--  dt 

y  -  z      X  -{-  y      .(•  +  ~  y  -  z      x  +  y  +  t      x  -\-  z  ^  t 


266  DIFFERENTIAL  EQUATIONS 

3.  Show  that  the  differential  equations  of  the  orthogonal  trajectories  (curves 
of  the  family  of  surfaces  F{x,  y,  z)  =  C  are  dx  :  dy.  dz  =  F^  :  F^  :  F^ .  Find  the  curves 
which  cut  the  following  families  of  surfaces  orthogonally  : 

(a)  aP-xP-  +  Vh'^  +  cV  =  C,         {^)  xyz  =  C,  (7)  y'^  =  Cxz, 

(5)  y  =  X  tan (2  +  C),  (e)  y  =  x  tan  Cz,         (f)  z  =  Cj-y. 

4.  Show  that  the  solution  of  dx:dy:dz=:  X :  Y :  Z,  where  X.  Y.  Z  are  linear 
expressions  in  x,  y,  z,  can  always  be  found  provided  a  certain  cubic  equation  cau 
be  solved. 

5.  Show  that  the  solutions  of  the  two  equations 

^  +  T{ax  +  by)  =  T„         '^  +  T{a^x  +  }/y)  =  T, , 

where  T,  T^,  T,  are  functions  of  t,  may  be  obtained  by  adding  the  equation  as 

~  {X  +  ly)  +  \T{x  +  ly)  =T^+IT.^ 

after  multiplying  one  by  /,  and  by  determining  \  as  a  root  of 
X-  -  («  +  '/)  X  +  ul/  -  a'b  =  0. 

6.  Solve:  (a)  t'^  +  2{x-y)  =  U         ^''^  +  x  +  5?/  =  <-\ 

dt  at 

(P)  tdx  =  {t-2x)  dt,  tdy  =  {tx  +  ty  +  2x-  t) dt, 

,    ,  Idx  mdy  ndz  dt 

(7) — = =  -  . 

mn  {y  —  z)       nl  {z  —  x)       Im  {x  —  y)        t 

7.  A  particle  moves  in  vacuo  in  a  vertical  plane  under  the  force  of  gravity  alone. 
Integrate.  Determine  the  constants  if  the  particle  starts  from  the  origin  with  a 
velocity  V  and  at  an  angle  of  a  degrees  with  the  horizontal  and  at  the  time  t  =  0. 

8.  Same  problem  as  in  Ex.  7  except  that  the  particle  moves  in  a  medium  which 
resists  proportionately  to  the  velocity  of  the  particle. 

9.  A  particle  moves  in  a  plane  about  a  center  (if  force  which  attracts  proportion- 
ally to  the  distance  from  the  center  and  to  the  mass  of  the  particle. 

10.  Same  as  Ex.  9  but  with  a  repulsive  force  instead  of  an  attracting  force. 

11.  A  particle  is  ^jrojected  X'a.rallel  to  a  line  toward  which  it  is  attracted  with 
a  force  i)roportional  to  the  distance  from  the  line. 

12.  Same  as  Ex.  11  except  that  the  force  is  inversely  proportional  to  the  square 
of  the  distance  and  onh'  the  patli  of  the  particle  is  wanted. 

13.  A  ])arti(;-le  is  attracted  toward  a  center  by  a  force  proportional  to  the  square 
of  the  distance.    Eind  the  orbit. 

14.  A  particle  is  jilaced  at  a  point  which  repels  with  a  constant  force  under 
which  the  particle  moves  away  to  a  distance  a  where  it  strikes  a  peg  and  is 
deflected  off  at  a  right  angle  with  undiminished  velocity.  Find  the  orbit  of  the 
subsequent  motion. 

15.  Show  tliat  (Mjmitions  (7)  may  be  written  in  the  form  (?rxF  =  0.  Eind  the 
condition  on  F  or  on  A'.  Y.  Z  that  tlic  integral  curves  have  (uthouonal  surfaces. 


MORE  THA^^   TWO  VARIABLES  267 

113.  Introduction  to  partial  differential  equations.  An  equation 
which  contains  a  dependent  variable,  two  or  more  independent  varia- 
bles, and  one  or  more  jmrtial  derivatives  of  the  dependent  variable 
with  respect  to  the  independent  variables  is  called  a,  jycrt  la  I  diffet-ent  led 
equation.    The  equation 

P (•^'  ^' ")  g^  +  ^ <^-^' '/'  '-^  ftj^^' ^'^'  ^'  ^'^'     ^ "" !7- '     '^^^/'   ^"^"^^ 

is  clearly  a  linear  partial  ditferential  equation  of  the  first  order  in  one 
dependent  and  two  independent  variables.  The  discussion  of  tliis  equa- 
tion preliminary  to  its  integration  may  be  carried  on  by  means  of  the 
concept  oi  phniar  elements,  and  the  discussion  will  immediately  suggest 
the  method  of  integration. 

When  any  point  (.r^^,  y^^,  z^  of  space  is  given,  the  coefficients  P,  Q,  R 
in  the  equation  take  on  definite  values  and  the  derivatives  p  and  q 
are  connected  by  a  linear  relation.  Xow  any  planar  clement  through 
(.r^,  y^,  z^  may  be  considered  as  specified  by  the  two  slopes  p  and  '/  ;  for 
it  is  an  infinitesimal  portion  of  the  plane  z  —  z^  =  p  (./•  —  x^^  -f-  y  (  y  —  y^ 
in  the  neighborhood  of  the  point.  This  plane  contains  the  line  or  lineal 
element  whose  direction  is 

dx  :  dii :  dz  =P:Q:R,  (15) 

because  the  substitution  of  P,  Q,  R  for  dx  =  x  —  x^^,  di/  =  y  —  y^, 
dz  —  z  —  z^  in  the  plane  gives  the  original  etpiation  Pjt  +  (iq  =  R. 
Hence  it  appears  tliat  the  planar  elements  defined  Ijy  (14),  of  which 
there  are  an  infinity  through  each  point  of  space,  are  so  related  tliat  all 
which  pass  through  a  given  point  of  space  pass  through  a  certain  line 
through  that  point,  namely  the  line  (lo). 

Now  the  prol)lem  of  integrating  the  etpiation  (14)  is  that  of  grouping 
the  planar  elements  Avhich  satisf}'  it  into  surfaces.  As  at  ea('h  point 
they  are  already  grouped  in  a  certain  Avay  by  the  lineal  elements  through 
which  they  pass,  it  is  first  advisable  to  group  these  lineal  elements  into 
curves  by  integrating  the  simultaneous  equations  (15).  The  integrals 
of  these  equations  are  the  curves  defined  by  two  families  of  surfaces 
F(x,  y,  z)  =  C^  and  G(x,  y,  z)  —  C.,.  These  curves  are  called  the  diarnr- 
tevlstlc  curves  or  merely  the  charaeterhtlcs  of  the  equation  (14j.  Through 
each  lineal  element  of  these  curves  there  pass  an  infinity  of  the  planar  ele- 
ments which  satisfy  (14j.  It  is  therefore  clear  that  if  these  curves  Ix'  in 
any  wise  grouped  into  surfaces,  the  planar  elements  of  the  surfaces  must 
satisfy  (14)  :  for  throiigh  each  point  of  the  surfaces  will  pass  one  of  the 
curves,  and  the  ])lanar  element  of  the  surface  at  that  point  must  there- 
fore pass  through  the  lineal  element  of  the  curve  and  hence  satisfy  (14). 


268  DIFFERENTIAL  EQUATIONS 

To  group  the  curves  F{x,  ;j,  z)  —  C\,  G (x,  ?/,  ,v)  =  C„  which  depend 
on  two  parameters  C^,  C,-,  into  a  surface,  it  is  merely  necessary  to  intro- 
duce some  functional  relation  (',  =  /"((; j  between  tlie  parameters  so 
that  when  one  of  them,  as  C\,  is  given,  the  other  is  determined,  and 
thus  a  particular  curve  of  the  family  is  iixed  by  one  parameter  alone 
and  will  sweep  out  a  surface  as  the  parameter  varies.  Hence  to  bit  eg  rate 
(1^),  first  integrate  (15)  and  then  vrlte 

6'(,r,  ,j,  z)  =  ^[F(x,  >/,  ,-)]      or     $(F,  G)  =  0,  (IG) 

where  <I>  denotes  any  arbitrary  function.  This  will  be  the  integral  of 
(14)  and  will  contain  an  arbitrary  function  $. 

As  an  example,  integrate  {y  —  z)p  +  {z  —  x)q  —  x  —  >j.    Here  the  equations 

ClX  (llJ  dZ  .  no  o  .-,  ^ 

= = give    x~  +  y-  +  ^-  =  C\,     .r  +  ?/  +  z  =  C.^ 

y  —  z      z  —  X      X  —  y 

as  the  two  integrals.    Hence  the  .solution  of  the  given  equation  is 

a;  +  2/  +  2  =  *  (x-  +  y-  +  Z-)     or     *  {x^  -{-  y-  +  z\  x  +  ij  -\-  z)  =  0, 

where  $  denotes  an  arbitrary  function.  The  arliitrarv  function  allows  a  solution 
to  be  determined  which  shall  pass  through  any  desired  curve;  for  if  the  carve  lie 
f{x.  v/,  z)  =  0,  (j{x,  y,  z)  —  0,  the  elimination  of  x,  y,  z  from  the  four  sinuUtaneous 
equations 

F{x,  y,  z)  =  C'l,         G  (x,  y,  z)  =  C.y,        /(x,  y,  z)  =  0,         (j  (x,  y,  z)  =  0 

will  express  the  condition  that  the  four  .surfaces  meet  in  a  point,  that  is,  that  the 
curve  given  by  the  first  two  will  cut  that  given  by  the  second  two  ;  and  this  elimi- 
nation will  determine  a  relation  between  the  two  parameters  C\  and  C,  which  will 
be  precisely  the  relation  to  expri'ss  the  fact  that  the  integral  curves  cut  the  givm 
curve  and  that  consequently  the  surface  of  integral  curves  passes  through  the  given 
curve.  Thus  in  tlu^  particular  case  here  considered,  suppose  the  solution  were  to 
pass  through  the  curve  i/  =  x-,  z  —  x  ;  then 

X- +  ?/- +  z- =  C',,         x  +  y  +  z-C.,,         y-x\         z  =  x 
give  2  X-  +  x*  =  Cj,         X-  +  2  X  =  C„, 

whence  {C'l  +  2  C._.  -  C\)-  +  8  C'|  -  24  <:\  -  10  (:\r.,  =  0. 

The  substitution  of  C\  =  x"  -f  //-  +  z-  ami  T'.,  =  x  +  y  +  z  in  tliis  eipiation  Mill 
giv(;  the  .solution  of  (y  —  z)p  +  (z  —  x)  ([  ^  x  —  y  which  passes  through  the  paral)ola 
//  =  X-.  z  =  X. 

114.  It  will  be  recalled  that  the  integral  of  an  ordinary  difiVi'- 
ential  e(piation  ./"(■''.//,//',•■■;  ,'/'"^j  =  ^^  <'^  tlie  nWi  order  contains  ii  con- 
stants, and  that  conversely  if  a  system  of  ctirves  in  the  plane,  say 
/•'(.'%//,  C'j,  •••,'"„)=  0,  contains  //  constants,  the  constants  may  li;' 
eliminated  from  the  c(|uati()n  and  its  first  ii  derivatives  with  respect 
to  ./•.  It  has  now  been  seen  that  tlie  integral  of  a  certain  jiai'tial 
diifcreiitial  (Mpiatlon   coniains  an  arl_)itrary  function,  and  it   miglit   be 


JNIOEE   THAX   T^V()  VAKIAIJLES  2(^9 

inferred  that  the  elimination  of  an  arbitrary  fiuu-tion  would  give 
rise  to  a  partial  dilferential  equation  of  the  first  order.  To  show 
this,  suppose  !■'(.'■,  >/,  a)  =  <J>[CV(./',   y,  ,t)].     Then 

7-;;  +  f:j>  =  <D' .  (/,-:.  +  rr^j>),         7-^  +  FTy  =  $' .  (rv;  +  r;:^) 
folloAV  from  j)artial  differentiation  with  respect  to  .?•  and  //;  and 

(JO-;  -  K'-~)p  +  (K''^  -  K^Oq  =  K''r  -  K'-:, 

is  a  partial  differential  equation  arising  from  the  elimination  of  <J>'. 
^Nlore  generally,  the  elimination  of  n  arbitrary  functions  will  give  rise 
to  an  e(piation  of  the  »th  order;  conversely  it  may  be  believed  that 
the  integration  of  such  an  e<|uation  Avould  introduce  n  arbitrary  func- 
tions in  the  general  solution. 

As  an  (.'xainplc,  eliiniiiate  from  z  =  ^(.ry)  +  ^  (x  +  y)  the  two  arbitrary  func- 
tions <i>  and  4'.    Tlie  lirst  iliffercntiation  gives 

p  —  <P'.y-\-^\         g  =  $' -x -f  ^',        p  —  q  =  {1/ —  x)^'. 

(~z  c'Z  c~z 

Now  differentiate  anain  and  let  r  =  —  .  «  = ,  t  --. Then 

CI-  c.rcy  c>j- 

r  —  s  =—  <i>'  +  (y  —  x) ^"  ■  y.         a  —  t  =  ^'  +  {y  —  x)<P"  ■  x. 

These  two  cijuatiiins  with  p  —  q  =  (y  —  x)'P'  nialvc  three  from  which 

x  +  y  r-z  c-z  c-z      x  +  y/cz      dz\ 

xr  -  {x  +  y)s  +  yt  = ^  (7)  -  q)    or   x  -,—  -  (.r  +  y)  -_-^  +  y  ---  = -  -- 

x  —  y  cX'  cxcy         cy      x  —  y\cz      cyl 

may  be  obtained  as  a  partial  differential  eijnation  of  the  second  order  free  from 
4>  and  ^.    The  general  integral  of  this  eijuatidn  would  be  z  =  ^{x.y)  +  4'  (.c  -f-  y). 

A  partial  differential  equation  may  represent  a  certain  definite  ty])e 
of  stirface.  For  instance  by  definition  a  conoidal  surface  is  a  surface 
generated  by  a  line  which  moves  })arallel  to  a  given  plane,  the  director 
plane,  and  ctits  a  given  line,  the  directrix.  If  the  director  ])lane  lie  taken 
as  ,-.■  =  0  and  the  directrix  be  the  ;v-axis,  the  equations  of  any  line  of 
the  surface  are 

::  =  f\,        y^r^;  with     r^  =  4>(^',) 

as  the  relation  which  picks  out  a  definite  family  of  the  lines  to  foi'in  a 
particukir  conoidal  surface.  Hence  ::  =  <!>(///./■)  may  l)e  regarded  as  the 
general  equation  of  a  conoidal  surface  of  which  ::  =  0  is  the  director 
plane  and  the  s-axis  the  dii'cctrix.  The  elimination  of  ^  gives  7/,/-  -f  y//  =  0 
as  the  differential  equation  of  any  such  conoidal  surface. 

Partial  differentiation  maybe  used  not  only  to  elimiiiate  arl)itrary  func- 
tions, but  to  eliminate  constants.  For  if  an  equation /'{,'•.  //,  .v,  (\,  ('„)  =  0 
contained  two  constants,  the  cfjuation  and  its  first  derivatives  with  respect 
to  ./■  and  II  would  yield  three  equations  from  which  the  constants  could 


270  DIFFERENTIAL   EQUATIONS 

be  eliminated,  leaving  a  partial  differential  equation  F{.r,  i/,  z,  p,  ^)  =  0 
of  the  first  order.  If  there  had  been  five  constants,  the  equation  with 
its  two  first  derivatives  and  its  three  second  derivatives  with  respect 
to  X  and  y  would  give  a  set  of  six  equations  from  which  the  constants 
could  be  eliminated,  leaving  a  differential  equation  of  the  second  order. 
And  so  on.  As  the  differential  equation  is  obtained  l)y  eliminating  the 
constants,  the  original  equation  will  be  a  solution  of  the  resulting  dif- 
ferential ecpiation. 

For  example,  eliminate  from  z  =  Ax^  +  2  Bxy  +  Cy-  +  Dx  +  Ey  the  five  con- 
stants.   The  two  first  and  three  second  derivatives  are 

p  =  2  Ax  -\--2.1hj  -\-  Z),       q  =  2  Bx  +  -2.Cy  +  E,       r  =  2  A,       s  =  2  B,       t  =  2  C. 

Hence  z  =—  I  rx'^  —  l  ty'^  —  sxy  4-  px  +  qy 

is  the  differential  efjnation  of  the  family  of  surfaces.  The  family  of  surfaces  do 
not  constitvite  the  ,<reneral  solution  of  the  e(iuation.  for  tliat  would  contain  two 
arbitrary  functions,  but  they  irive  what  is  called  a  rouiplete  solution.  If  there  had 
been  f)nly  three  or  four  constants,  the  elimination  would  have  led  to  a  differential 
equation  of  the  second  order  which  need  have  contained  only  one  or  two  of  the 
second  derivatives  instead  of  all  three  ;  it  would  also  have  been  possible  to  find  three 
or  two  simultaneous  partial  differential  equations  by  differentiating  in  different  ways. 

115.    If     f(.r,  I/,  z,  C\,  Q  =  0      and      F(.r,  >/,  z,  p,  y)  =  0  (17) 

are  two  equations  of  which  the  second  is  obtained  hy  the  elimination  of 
the  two  constants  fi'om  the  fii'st,  tlie  first  is  said  to  be  the  com jiU-fc  soJii- 
tlon  of  the  second.  That  is,  any  equati(^]r  Avliicli  contains  two  distinct 
arl)itrary  constants  and  wliicli  satisfies  a  jiartial  differential  equation  of 
the  first  order  is  said  to  In-  a  complete  solution  of  the  differential  equa- 
tion. .V  com[)lete  solution  has  an  interesting  goometric  interpretation. 
The  differential  equation  F=0  di.'fines  a  stu'ics  of  }ilanar  elements 
tln'f)ugh  each  ])oint  of  s])ace.  So  does /(./•,  //,  ;.•,  C^,  r ' j  =  0.  For  the 
tangent  plane  is  given  l.)y 


as  the  condition  that  C^  and  (".,  sliall  be  so  I'clated  tliat  tlie  surface 
passes  througli  (./■.,.  //.^.  z^).  As  tlicrc  is  only  tliis  one  i-olatioii  lictweon 
the  two  arbitrai'v  constants,  there  is  a  avIioIo  scries  of  ])l;inar  elements 
through  the  point.  As  /(.r,  //.  z.  C'^.  r'j  =  0  satisfies  the  differential  t^qua- 
tion,  the  planar  elements  defined  by  it  are  those  defim^d  by  tlie  diffei'en- 
tial  equation.  Thus  a  conq)lete  soluticjii  establishes  an  arrangement  of 
the  ]»lanar  elements  defined  by  tlu'  differential  ('(piation  u])on  a  family 
of  surfaces  dependent  upon  two  arbitrary  constants  of  integration. 


MOKE  THAN  TWO  VARIABLES  271 

From  the  idea  of  a  solution  of  a  partial  differential  equation  of  the 
first  order  as  a  sui'faee  })ieeed  together  from  planar  elements  "whieh 
satisfy  the  equation,  it  appears  that  the  envelo})e  (p.  140)  of  any  family 
of  solutions  will  itself  be  a  solution ;  for  each  point  of  the  envelope  is 
a  point  of  tangeney  with  some  one  of  the  solutions  of  the  family,  and 
the  planar  element  of  the  envelope  at  that  point  is  identical  with  the 
planar  element  of  the  solution  and  hence  satisfies  the  differential  equa- 
tion. This  ohxcrrailon  aJhurs  tlie  rjeimral  solution  to  he  itcicrmlncA  fnnii 
any  complete  solution.  For  if  in  /(j-,  y,  z,  C\,  ('.,)  =  0  any  relation 
C.,  =  4>(C'j)  is  introduced  between  the  two  arbitrary  constants,  there 
arises  a  family  depending  on  one  parameter,  and  the  envelope  of  the 
family  is  found  by  eliminating  C^  from  the  three  equations 

cf        cl^  df 

As  the  relation  C,  =  <&(Cj)  contains  an  arbitrary  function  4>,  the  result 
of  the  elimination  may  be  considered  as  containing  an  arbitrary  func- 
tion even  though  it  is  generally  im})Ossible  to  carry  out  the  elimination 
except  in  the  case  where  4>  has  been  assigned  and  is  tlierefore  no  longer 
arbitrary. 

A  family  of  surfaces  fCr,  y,  ':,  C^,  C,)  =  0  depending  on  two  param- 
eters may  also  have  an  envelope  (p.  139).  This  is  found  by  eliminat- 
ing C\  and  C„  from  the  three  e(]uations 

cf  cf 

f(,r,y,z,C\,C^  =  0,  ^  =  0,  ^=0. 

This  surface  is  tangviit  to  all  the  sui'faces  in  the  cCJinph'te  solution. 
This  envelope  is  called  the  slnyuhtr  solution  of  the  partial  differential 
equation.  As  in  the  case  of  oi-dinary  differential  ecpiations  (§  101),  the 
singular  solution  may  be  obtained  directly  from  the  equation;*  it  is 
merely  necessary  to  eliminate  y^  and  q  from  the  tliree  equations 

F(.,y,.,y,v)  =  0,  -=0,  ^  =  0. 

The  last  two  equations  express  the  faet  that  F( p,  if)  =  0  regarded  as 
a  function  of  y/  and  y  should  ha\e  a  double  point  (^  57).  A  reference 
to  ^  ()7  will  bring  out  another  point,  namely,  that  not  only  are  all  tlie 
surfaces  represented  by  tlie  complete  solution  tangent  to  the  singular 
solution,  but  so  is  any  surface  whidi  is  represented  l)y  the  general 
solution. 

*  It  is  li!U'(lly  iiecossary  to  point  out  thf  fact  that,  as  in  the  case  of  ordinary  equations, 
extraneous  factors  may  arise  in  the  eliniiiiatioii,  wliether  of  <  \,  ''/o  or  of  p,  q. 


272  DIFFEKENTIAL  EQUATIONS 

EXERCISES 

1.  Integrate  tliese  linear  e(iuations: 

{a)  xzp  +  yzq  =  ry,  (/3)  a  {p  +  q)  =  z,  (7)  i-p  +  y-q  =  z-, 

(5)  -  ?/p  +  J'y  +  1  +  z-  =  0,  (e)  ?/p  -  xq  =  X-  -  y-,         (i")  (X  4-  2)i>  =  y, 
(n)  x^p  -  xyq  +  y-  =  0,  {0}  (a  -  x)p  +  (b-y)q  =  c-  z, 

{i)  p  tan X  +  q  tan  y  =  tan  z,  (/c)   {y-  +  z-  —  x-)_p  —  2 /^r/  +  2 /z  =  0. 

2.  Determine  the  integrals  of  the  iireceding  eejuation.s  to  pa.ss  through  the  curves  : 

for     (a)  /-  +  2J-  =  \,  z  -0,  for     (/3)  y  =  0.  x  =  z, 

for     (7)   ;/ =  2x,  z  =  1,  for     {e)  x  =  z,  y  =  z. 

3.  Show  analytically  that  if  F{x,  y,  z)  =  (\  is  a  solution  of  (15),  it  is  a  solution 
of  (14).  State  precisely  what  is  meant  by  a  solution  of  a  partial  differential  equa- 
tion,  that  is,  by  the  statement  that  F(x,  y,  z)  —  C\  satisfies  the  equation.  Show  that 
the  equations 

p  ^1  4-  f^  —  ==  A'     and     P—  +  (/--  +  I!  --  =  0 
cx  iy  ex  cy  Iz 

are  eciuivalent  and  state  what  this  means.  Show  that  if  F  =  i\  and  (i  ~  ('„  are 
two  solutions,  then  F  —  ^(G)  is  a  solution,  and  show  ronversely  that  a  functional 
relation  must  exist  between  any  two  solutions  (sec  §  (;2). 

4.  Generalize  the  work  in  the  text  along  tlie  analytic  liiies  of  Ex.  3  to  estal)- 
li.sh  the  rules  for  integrating  a  linear  equation  in  one  dependent  and  four  or  n 
independent  variables.    In  particular  show  that  the  integral  of 

cz                          cz                                               c/x,                   d.r„          (h 
P-^^—  +  ■■■  +  Pn^r-  =  Pn+i     depends  on     ----!•  =  ...=         = , 

fXj  fX„  /j  P„  Iu+1 

and  that  if  F^  —  (\.  ■  ■  ■.  F„  ^  ('„  are  n  integrals  of  the  siniultaneous  system,  the 
integral  of  the  partial  differential  e(juation  is  4>(-/'\,  •  ■  ■,  F„)  =  0. 

t     ,  ,    ,      cu  cii         cu 

o.  Integrate  :  (a)  x \-  y       ~\-  z  ---  =  xyz, 

cx  cy         cZ 

(/3)    (y  +  2  +   i')  T^  +   (2  +   «   +  X)  '-'   +   (,,   +  X  +  //)  '"   :^  X  +   //  +  Z. 

CX  cy  cz 

6.  Interpret  the  general  equation  of  the  lirst  order  F(x.  y.  z.  p.  7)  =  0  as  detcr- 
iniuing  at  each  point  (x,,.  2/,,,  2,,)  of  i^pace  a  series  of  j)laiiar  eh-nicnts  tangent  to  a 
certain  cone,  namely,  tlie  cone  found  by  eliminating  ;>  and  7  from  ilic  iliree  simul- 
taneous equations 

^'K'  ^o:  ^0'  P^  'l)  =  0-         (•'■  -  •'",.)  i'  +  <■'/  -  //o)  7  =  2  -  ^u' 

cF  cF      ^ 

(x-x„)-_--  -(//-  y^)  _  -  =0. 

cq  cp 

7.  Eliminate  the  arbitrary  functions: 

{a)  J-  +  y  +  ^  =  <f'  (X-  +  .'/-  +  Z-),  (;3)   *  (X-'  +  y-\  z  -  sy)  =  0, 

(7)  z  -^{s  -{-  y)  +  ^(x  —  y),  (5)  z  =  c""4>(x  —  y). 

(e  )  z  =  //-  +  2  <l>  (X-  1  +  log  ,y),  (X)  <}>  (■'■ ,    ■'- ,   ~)   =  0. 

\ij     z     x/ 


I\IOrvE   THAN   TWO  VAKIAT-LES  278 

8.  Find  tlic  differt'iitiiil  (Mjuatioiis  of  these  types  of  surfaces: 

(a)  cylinders  witli  ^a'uerators  parallel  to  tlie  line  x  =  az,  y  =  bz, 

(13)  conical  surfaces  with  vertex  at  {a.  h.  <"), 

(7)  surfaces  of  revolution  about  the  line  x  :  ij  -.z  ~  a:h  -.c. 

9.  Eliminate  the  constants  from  these  eijuations : 

(a)  z  =  is  +  u)  (g  +  /,^,  (^)  a  (j-2  +  ,f-)  +  ^2  =  1, 

(7)   (X  -  nf  +  (y  -  hf  +  (2  -  cf-  =  1,        (5)   (X  -  a)-  +  (y  -  '')-  +  (z  -  c)2  =  d^, 
(e )  .l.r2  +  /;x^/  +  Cif-  +  JJjz  +  7;//z  ^^  Z-. 

10.  Show  jjeometrically  and  analytically  that  i'"(/,  y,  z)  +  aG  {x,  y,  z)  —b  in  ^ 
complete  solution  of  the  linear  eijuation. 

11.  How  many  constants  occur  in  the  complete  solution  of  the  ecjuation  of  the 
third,  fourth,  or  ?ith  order? 

12.  Discuss  tlie  cumphte.  general,  and  simrular  solutions  of  an  e(]uation  of  the 
first  order  F{x.  y.  z,  »,  (/',.,  i/,,  u'._)  =  0  with  tliree  imlependent  variables. 

13.  Show  that  the  planes  z  =  ax  +  by  -{-  ('.  where  a  and  '/  are  connected  by  the 
relation  F{fi.  b)  =  0.  are  complete  solutions  of  the  eiiuation  F{p.  '/)  =  0.    Integrate  : 

(a)  p>i  =  1,         {13}  q=  [r  +  1,  (7)  p-  +  q-  =  "'-- 

( 5 )  P'l  =  A-,  {e)  k  lo-  q  +  p  =  0,  ( j-)   3i>2  _  2  ,f  =  ipq^ 

and  determine  also  the  siuLrular  solutions. 

14.  Note  that  a  simple  chancre  of  variable  will  often  reduce  an  equation  to  the 
type  of  Ex.  13.    Thus  the  eijuations 

f(^,'^)=0,       Fixp,q)  =  0,       ^(f.  f)=0. 

with  z  =  c~\  X  =  €■'''.         z  =  C'',  X  —  (:■''.  y  =  e'J', 

take  a  simpler  form.    Infeurate  and  determine  the  singular  .solutions: 

((t)  q  ^  z  +  p£,         iii)  x-p-  +  //-'/-  =  '^-,  (7)  2  =  Vq- 

(5)   7  =  2  yp-.  (e)    {p  -  yf  +  (7  -  xf  =  1,         (D   z  =  /'"V"- 

15.  What  is  the  obvious  complete  .'solution  of  the  extended  C'lairaut  eipiation 
z  =  xp  +  y(j  +  fip.  q)  ?    l)iscu.<s  the  singular  solution.    Integrate  the  ecjuations : 

(a)  z  =  xp  +  yq  +  ■\'p)-  +  q-  +  1,  (p)  z  =  xp  +  y/  +  (p  +_q)-, 

(7)  z  =xp  +  yq  +  pq,  (5)  z  =  xp  +  yq  -  2  ^'ptq. 

116.  Types  of  partial  differential  equations.  In  addition  to  the 
liiu'tii-  eijuiitioii  ;uid  the  typt-s  of  Exs.  13-15  altove,  there  are  several 
types  which  shoiihl  be  mentioned.  Of  tliese  the  first  is  tlte  general 
e<[}i<it[nn  tif  tin'  jir><t  arih-r.  If  /•"('.'■,  y,  z,  p,  q)  =  0  is  tlie  given  equation 
tmd  if  a  second  e(|u;ition  <!>(./■.  //,  '-.p.  y,  ")  =  0,  wliieh  holds  simultane- 
ously vritli  the  first  tiiid  contains  an  arl)itrary  constant  e;in  l)e  found, 
the  two  eijutitions  mav  l»e  solved  together  for  the  vtilues  of  y>  and  y,  and 
the  results  m;iy  he  stiltstituted  in  tlie  relation  dz  =  p'Tj'  +  y"'.'/  f"  yi'^'^*  'i' 
tottil  differential  eqtiation  (jf  wdiicli  the  integrtil  will  contain  the  eon- 
stant  (I.  aiid  a  second  constant  of  integration  h.    This  integrtd  will  then 


274 


DIFFERENTIAL  EQUATIONS 


be  a  complete  integral  of  the  given  equation  ;  the  general  integral  may 
then  be  obtained  by  (18)  of  §  115.    This  is  known  as  Clmvplt^s  metJtod. 
To  find  a  relation  4>  =  0  differentiate  the  two  equations 

F{x,  y,  z,  2h  y)  =  0,  ^{x,  >j,  z,  p,  q,  a)  =  0 

with  respect  to  x  and  //  and  use  the  relation  that  dz  be  exact. 


(19) 


(l.i-  a.r 

^^  +  <!' 7>  +  ^;  y- +  $  /  =  0, 

(IX  (Ix 

,  dp  ,  (hi 


dii       dx  ' 


Multiply  by  the  quantities  on  the  right  and  add.    Then 

(K+pK)  ^  +  (f; + ,K,  -  -  ^;  -  -  ^;  ^  -  (j,f; + ,,f;,  ^  =  o.  (20) 

Now  this  is  a  linear  equation  for  <I>  and  is  equivalent  to 

dp        ^        d<I        ^    dx    ^    dif    ^  dz  ^  (I^ 

K  +i>K     K  + 1^^^     -  K     -  K     -  O'K  +  'iK)      o  '   ^-  ^ 

Any  integral  of  tliis  system  containing  p  or  q  and  a  will  do  for  4>,  and 
the  sim})lest  integral  will  naturally  be  chosen. 

As   an   example   take   zp  (x  +  ?/)  +  _2)(r/ —  j<)  —  2- =  0.     Then    Charpit's   eqna- 
tions  are 

dp  _  '^^  _  ^^■'' 

-  zji  +  p2  (X  +  y)       zp  -2zq  +  pxi  {X  +  2/)       2p-  q-  z(z-\-  y) 
_   (ly   _  dz 

—  p       2  p'^  —  2  p>q  —  pz  (x  +  7/) 

How  to  combine  these  so  as  to  get  a  sohition  is  not  very  clear.    Suppose  the  sub- 
stitution z  =  e^',  p  —  e^'p',  q  =  c~V/'  l)e  made  in  the  equation.   Then 

i>'  (-c  +  //)  +  p'  ('/  -  p')  -1  =  0 
is  the  new  ecination.    Por  this  Chai'pit's  sinniltancous  system  is 
dp'  _  dq'  _  dx  _    dy    ^  dz 

p'        p'       2  p'  —  (['  —  (x  +  y)       —  p'       -Ip"^  —  2pq  —  p'  (x  +  y) 

The  first  two  e(iuati()iis  ^ive  at  once  the  solution  dp'  =  d(['  or  (/'  =  !>'  +  «•    Solving 

p'  (X  +  y)  +  p'  ('/'  —  P')  —1  =  0     and     f/'  =  p'  +  «, 

dx  +  dy 


P 


1 

a  +  X  +  y 


1 

((  +  X  +  y 


+  ", 


dz'  = 


a  +  X  +  y 


+  ady. 


MOliE  THAX  TWO  VAEIABLES  275 

Then      z  =  log  (a  +  x  +  y)  +  ay  +  b     or     \ogz  =  log  (a  +  x  +  y)  +  ay  +  b 

is  a  complete  solution  of  the  given  etiuation.  This  will  determine  the  general 
integral  by  eliminating  a  between  the  three  equations 

z  =  e«2^  +  6(tt  +  x  +  ?/),        b=f(a),        0  =  {y +f\a))(a  +  x  +  y)  +  1, 

where  f{a)  denotes  an  arbitrary  function.  The  rules  for  determining  the  singular 
solution  give  z  —  0;  but  it  is  clear  that  the  surfaces  in  tlie  complete  solution  can- 
not be  tangent  to  the  plane  z  =  0  and  hence  the  result  z  =--  0  must  be  not  a  singular 
solution  but  an  extraneous  factor.    There  is  no  singular  solution. 

The  method  of  solving  a  partial  diii'erential  e(|uatioii  of  higher  order 
than  the  first  is  to  reduce  it  first  to  an  e(}uation  of  the  iirst  order  and 
then  to  complete  the  integration.  Frecjuently  the  form  of  the  equation 
will  suggest  some  method  easily  applied.  For  instance,  if  the  deriva- 
tives of  lower  order  corresponding  to  one  of  the  independent  variables 
are  absent,  an  integration  may  be  performed  as  if  the  equation  were 
an  ordinary  equation  with  that  variable  constant,  and  the  constant  of 
integration  may  be  taken  as  a  function  of  that  variable.  Sometimes  a 
change  of  variable  or  an  interchange  of  one  of  the  independent  variables 
with  the  dependent  variable  will  sim})lify  the  equation.  In  general  the 
solver  is  left  mainly  to  his  own  devices.  Two  special  methods  will  be 
mentioned  below. 

117.  If  the  equation  is  Imcar  with  constant  coefficients  and  all  the 
derivatives  are  of  the  same  order,  the  equation  is 

{%D';  -f  njyr'i>,  +  ■  •  •  +  "„-i />„./>;-'  +  ''.^;)-  =  ^^C'-.  !/)■    (22) 

Methods  like  those  of  §  95  niay  be  a])plied.    Factor  the  e(p;ation. 

a^{D,  -  a^D^)  {D,_.  -  a^^;)  •  •  •  (7),.  -  ajl,)  r:  =  E  (.r,  y).  (22') 

Then  the  lupiation  is  reduced  to  a  succession  of  equations 

each  of  which  is  linear  of  the  fii'st  order  (and  with  constant  coefficients). 
Short  cuts  analogous  to  those  previously  given  may  be  developed,  but 
will  not  be  given.  If  the  derivatives  are  not  all  of  the  same  order  but 
the  polynomial  can  be  factored  into  linear  factors,  the  same  method  Avill 
api)ly.  For  those  interested,  tlie  several  exercises  given  belo\v  Avill  serve 
as  a  syno})sis  for  dealing  Avith  these  types  of  equation. 
There  is  one  equation  of  the  second  order,*  namely 

V'-  if-       c.r       cij-       drc-  ' 

*  Tliis  is  one  of  the  iniportiiiit  differential  equations  of  physics  ;  other  important  equa- 
tions and  methods  of  treating  them  are  discussed  in  Chap.  XX. 


276  DIFFEKEXTIAL  EQUATIONS 

wliicli  occurs  constantly  in  the  discussion  of  waves  and  wliich  has  there- 
fore the  name  of  tlie  tcave  e'lttatlon.  The  solution  may  be  written  down 
by  inspection.    For  try  tlie  form 

n  (.r,  I/,  r:,  f)  =  F{ax  +  In,  +  cz  -  Vt)  +  G{ax  +  hj  +  cz  +  lY).    (24) 

Substitution  in  the  e(|uation  shows  that  tliis  is  a  solution  if  the  relation 
(('-  +  Ir  +  c-  =  1  holds,  no  matter  what  functions  F  and  (^  uiay  l)e.  Note 
that  the  equation 

a.r  +  ],;j  +  cz  —  T7  =  0,  n-  +  //-  +  c"  =  1, 

is  the  equation  of  a  jilane  at  a  ])erpendicul;ii'  distance  17  from  the  origin 
along  the  direction  whose  cosines  are  a,  l>,  c  \i  t  denotes  the  time  and 
if  the  plane  moves  away  from  the  origin  with  a  velocity  V,  the  function 
F{(ix  +  by  +  ez  —  17)  =  i^(0)  remains  constant ;  and  if  G  =  0,  the  value 
of  u  will  remain  constant.  Thus  a  =  F  represents  a  phenonu^non  wliich 
is  constant  over  a  })lanp  and  retreats  with  a  velocity  \',  that  is.  a  plane 
wave.  In  a  similar  manner  u  =  G  re})resents  a  plane  wave  a])]ii-(jaching 
the  origin.  The  general  s(jlution  of  (2.S)  therefore  re])resents  the  super- 
position of  an  advancing  and  a  retreating  })lane  Avave. 

To  Monp:e  is  due  a  metliod  sometimes  useful  in  treatiiiic  ilifferential  e(|uation.s 
of  the  .second  order  linear  in  the  derivatives  r,  .s,  t  ;  it  is  known  as  Mumjc'.s  vuthod. 

Let  Ur  -t-  i^'.s  ^  Tt  =  V  (25) 

be  the  e<ination,  where  7i,  S.  T.  V  are  functions  of  the  variables  and  the  derivatives 
23  and  q.    From  the  given  equation  and 

dp  =  rdx  +  sdy.         dq  =  sdx  +  tdy, 

the  elimination  of  r  and  t  gives  the  e(iuation 

s{Edy-  -  Sdxdy  +  Tdx-)  -  {lldydp  +  Tdxdq  -  Vdxdy)  =  0, 

and  this  will  surely  be  satisfied  if  the  two  equations 

lldy-^  -  Sdxdy  +  Tdx-  =  0,         lUlydp  +  Tdxdq  -  Vdxdy  =  0  (25') 

can  be  .satisfied  simultaneously.    The  first  may  lie  factored  as 

dy  — /^  (x.  y.  z,  p,  q)dx  =  0,         dy  —  /„  (x.  y.  z.  p.  '/)'/.'•  =  0.  (2(;) 

The  pi'dblem  then  is  reduced  to  integrating  the  system  ^insisting  iif  one  of  these  f;u-- 
tors  with  (25')  iuu\dz  =  j>dx-i-qdy.  that  is,  asysteninf  three  total  ditrei'cntial  equations. 
If  two  indejiendent  solutions  of  this  system  can  lie  found,  as 

*/i  (x.  y.  z.  p.  q)  =  C'l,  u..  (x.  y.  z.  p.  q)  =  ('.., 

then  7q  —  4> '//.,)  is  a  lii'st  oi'  intermediary  integral  of  tlie  uiven  equation,  tlie  general 
integral  of  which  uia\'  be  found  by  integrating  tliis  equation  of  the  lirsl  oriler.  If 
tiie  two  factors  are  di.-tim-t.  it  may  happen  that  the  two  systems  which  arise  ni;i>' 
both  be  integrated.  Tlien  two  fir>t  integrals  /(,  :--  'f' (/'.,)  and  r,  =-  ^'  (r„)  will  be  found. 
and  instead  of  intcgi'atiug  one  of  these  equations  it  may  be  better  to  sohc  both  for 
p  and  (/  and  to  substitute  in  tlie  ex]iression  dz  =z  pdx  +  qdy  and  integrate.  When, 
however,  it  is  not  iios>ible  to  find  even  one  first  inteural.  .Moiiai-'s  method  fails. 


MORE  TIIAX  TWO  VARIABLES  277 

As  an  example  take  (x  +  y)  [r  —  t)  =—  ijK    The  equations  are 

{x  +  y)  dy'^  —  {£  +  y)  dx-  =  0     or     dy  —  dx  =  0,         dy  -\-  dx  =  0 
and  (X  +  y)dydp  —  (x  +  y)dxdq  +  ipdxdy  =  0.  (A) 

Now  the  equation  dy  —  (Zx  =  0  may  be  integrated  at  once  to  give  y  =  x  +  (,\.    Tlie 
second  eijuation  (A)  tlien  takes  tlie  form 

2  xtZp  +  ipdx  -  2  xdq  +  C^  (dp  —  dq)  =  0  ; 
but  as  dz  =  pdx  +  qdy  =  (p  +  q)dx  in  this  case,  we  liave  by  combination 

2  {xdp  +  pdx)  —  2  {xdq  +  qdx)  +  C'^  ((Zj)  —  (Z(/)  +  2  tZ^  =  0 
or  (2  X  +  r ,)  ( ?;  -  V)  +  2  .-  =  ( ',     ( n-     (x  +  y)  (p  -  q)  +  2  z  ■=  C,. 

Hence  (x  +  y)  {])  —  q)  +  2z  =  ^{y  —  x)  (27) 

is  a  lirst  integral.    'I'his  is  linear  and  may  be  integrated  by 

cZx  di/  dz  -,  dx  dz 

or    X  +  y 


x+y  x  +  y      $  (//  -  X)  -  2  2  K\      4>  (iv",  _  2  x)  -  2  z 

This  equation  is  an  ordinary  linear  equation  in  z  and  x.    The  integration  gives 

I\\ze Jh  =    I  c /''i *  ( K\  —  2  x)  dx  +  Ju . 
Hence        (x -^  ij)  ze  +  <' —    I  L'^^-i>{K^  — 2x)dx  =  K„  = -^  [K  ^)  =  ^  {x  +  y) 

is  the  general  integral  of  the  given  equation  Avhen  K^  has  beoi  replaced  by  x  +  ?/ 
after  integration,  —  an  integration  which  cannot  be  perfoi-med  until  <I>  is  given. 

'J'he  other  nu'thod  of  solution  would  l)e  to  use  also  tln'  second  system  containing 
dy  +  dx  =  0  instead  of  dy  —  dx  =  0.  Thus  in  addition  to  the  lirst  integral  (27)  a 
second  intermediary  inti'gral  might  be  sought.  The  substitution  of  dy  +  dx  =  0, 
y  +  X  =  (\  in  (A)  gixcs  (\  [dp  +  dq)  +  4p)dx  =  0.  'I'his  eipiation  is  not  int-egrable, 
because  d]>  +  (Z7  is  a  perfect  ditTerential  and  ptZx  is  not.  The  combination  with 
dz  =  pdx  +  qdy  =  {p  —  q)dx  does  not  improve  matters.  Hence  it  is  impossible  to 
determine  a  second  intermediary  integriil,  i^nd  the  method  of  completing  the 
solution  by  integrating  (27)  is  the  only  available  method. 

Take  the  equation  j>.s  —  qr  =  0.    Here  S  =  p,  1!  =  —  (/,  T  ~V  =  0.    Tlien 

—  qdy-  —  pidxdy  —  0     or     dy  =  0,     pdx  +  qdy  —  0     and     —  (/dydp  =  0 

are  the  ('(juations  to  work  with.  The  system  dy  —  0,  qdydp  =  0.  dz  =  pdx  +  qdy, 
and  the  system  pdx  +  (/dy  =  0.  tjdydp  =  0.  ilz  ~  ])dx  +  qdy  are  not  very  satisfactory 
for  obtaining  an  iiUermcdiary  integral  k^  =  *('(.,),  although  ]>  —  *(-)  is  an  ol)vions 
solution  of  the  lirst  set.  It  is  better  t(.)  use  a  method  adapted  to  this  spi'cial 
iMjuation.    Ntjte  that 

^-(")=^'-'/"'^    attd     ^(''):=0    gives    ?  =  /(,). 
''=-p)^      then      ^  =-/(,) 
and  X  =  -  ^f(y)  dy  +  ^I'  (-)  r^  *  {y)  +  <i'  (z). 


IX  \p'  ]:-  ex  \i 

P.y  (11),  p.  124,  ''-  =-  ('A  ■     then     '^=-f{y) 

p  \cy'-:  cy 


278  DIFFERENTIAL  EQUATIONS 

EXERCISES 

1.  Integrate  these  equations  and  discuss  the  singular  sohition : 

{a)  pi  +  q^  =  2x,            (^)   (p^  +  q^)  j  =  py^  (^)   (^  +  ^)  (^3.  4.  ^^z)  =  1, 

(5)  pg  =px  +  (/?/,            (e)  p'^  +  ^2  =  X  +  2/,  (f)  xp2- 2zp  +  j-i/ =  0, 

(^)   y2  ^  Z^  (p  -  fy),           (^)   q  ip-h  +  q')  =  h  (0  i^  (1  +  ^r)  =  </  (^  -  c), 

( »c )  xp  (1  +  r/)  =  (/z,          (X)   ij'  (p-^  -  1)  =  x2p2,  (^)  Z--2  (pi  +  r/  +  1)  =  c^ 

( »/)  p  =  (z  +  y'i)-,             (0)  pz  =  l+  q-,        (tt)  z-pq  =  0,        (p)  r/  =  xp  +  p2. 

2.  Show  that  the  rule  for  the  type  of  Ex.  13,  p.  273,  can  be  deduced  by  Charpit's 
method.    How  about  the  generalized  Clairaut  form  of  Ex.  15  ? 

3.  (a)  Eor  the  solution  of  the  type/](x,  p)  —-f.,{ij^  (/),  the  rule  is  :  Set 
and  solve  f or  p  and  ry  asp  =  g■^(x^  a),  q  =  y.,{y,  a)  ;  the  complete  solution  is 

^  =  f  (Jii-^-  «)'^-«  +  fooiy-,  «)''y  +  '^• 

(/3)  Eor  the  type  F{z,  p,  q)  =  0  the  rule  is :  Set  X  =  x  +  ay,  solve 

(dz         dz  \                dz                                         c     dz 
z,  ,  a tor     =  d)  [z,  a),    and  let      |  —f(z,  a) ; 

the  complete  snlution  is  x  +  ay  +  h  =f{z,  a).  Discuss  these  rules  in  the  light  of 
Charpifs  method.  Establish  a  rule  for  the  type  F{x  +  y,  p.  q)  =  0.  Is  there  any 
advantage  in  using  the  rules  over  the  use  of  tlie  general  method  ?  Assort  the  exam- 
ples of  Ex.  1  according  to  these  rules  as  far  as  possible. 

4.  What  is  obtainable  for  partial  differential  equations  out  of  any  characteristics 
of  homogeneity  that  may  be  present '? 

5.  By  differentiating  p  —f{x.  y,  z.  q)  successively  with  respect  to  x  and  y  show 
that  the  expansion  of  the  solution  by  Taylor's  Foruuila  about  the  point  (x^.  y^J.  z,j) 
may  be  found  if  the  successive  derivatives  with  respect  to  y  alone, 

cz  c-z  c^z  c"z 

cy  cy-  cy^  cy" 

are  assigned  arbitrary  values  at  tliat  point.  Note  that  this  arbitrariness  allows  the 
solution  to  be  passed  througli  any  curve  through  {x^,,  y^^,  z^^)  in  the  plane  x  =  x^. 

6.  Show  that  F{x,  y,  z.  p,  q)  =  0  satisfies  Charpifs  e(}uations 

au  =  ^^^=^^  = '^ =        '^^'        =        '^'^       ,  (28) 

-K    -K    -O'K  +  'iK)    K  +  pK    K  +  'iK 

wliere  u  is  an  auxiliary  variable  introduced  for  symmetry.  Show  that  the  first 
three  ecjuations  are  the  differential  equations  of  the  lineal  elements  of  the  cones  of 
Ex.  6,  p.  272.  The  integrals  of  (28)  therefore  define  a  system  of  curves  whicli  have 
a  planar  element  of  the  ecjuation  F  =  0  passing  through  each  of  their  lineal  tan- 
gential elements.  If  tlie  e(]uatioiis  be  integrated  and  the  results  be  solved  f(ir  tlie 
variables,  and  if  the  constants  Ije  so  determined  as  to  specify  one  particular  curve 
with  the  initial  conditions  x,,.  ?/,,,  z,j,  p^,  7,,.  then 

x  =  x(u,  Xq,  ?/o,2:o,Po,  (y,j),     y  =  y{---),z  =  z{---),    p=p{---),     q^q{---). 


UOllE  THAN  TWO  VARIABLES  279 

Note  that,  aloiis;  tlie  curve,  q  =/{]))  and  that  consequently  the  planar  elements 
just  mentioned  must  lie  upon  a  developable  surface  containing  the  curve  (§  G7).  The 
curve  and  the  planar  elements  along  it  are  called  a  characteristic  and  a  dtaracttrldic 
strip  of  the  given  differential  etiuation.  In  the  case  of  the  linear  equation  the 
characteristic  curves  afforded  the  integration  and  any  planar  element  through 
their  lineal  tangential  elements  satisfied  the  equation  ;  but  here  it  is  only  those 
planar  elements  which  constitute  the  characteristic  strip  that  satisfy  the  equation. 
What  the  complete  integral  does  is  to  piece  the  characteristic  strips  into  a  family 
of  surfaces  dependent  on  two  parameters. 

7.  By  simple  devices  integrate  the  e(iuations.    Check  the  answers: 

cj-  cif  cscij      y 

{5)  s  +  pf(f)  =  g(y),  {e)ar  =  xy.  {^)  xr  =  {n  -  l)p. 

8.  Integrate  these  equations  by  the  method  of  factoring: 

(a)   (Ui  -  a^D^ )z^0.         (^)   {!),  -  lJ,f  z  =  0.         (7)   ^7),!);  -  D^) z  =  0, 
( 5 )   {Ui  +  3  I)J)y  +  2  7J;-)  2  =  J  +  y.         ( e )   (i>f.  -  7/,  /A,  -  0  7>; )  z^xy, 
( r)   (^I-  -  7>^  -  3 7>,,  +  3 l),j) z  =  0,  (,,)   (1)1  -ljl  +  -2D,+  l)z  =  e--. 

9.  Prove  the  operational  equations  : 

(a)  £«-'A/0  (//)  =  (1  +  axl)„  +  '.  a-x-IJ;  +  ■  •  •)  0  (.?/)  =  <P(y  +  ax), 
(/3) ~  0  =  e"-^,/  -     0  =  e"-!',,  ^{y)  =  4>(y  +  ax), 

J  J  J.  —   aJ)i,  J),r 

(7)  7 ^—r  ^'  (•'••  y)  =  ^"'■^'."  r  '^'~  ''^^vT?  (f.  y)  d^  =  ril  (t,  y  +  ax-a^)dk. 

Lj.  —  alJy  -J  J 

10.  Prove  that  if  [(7)^  -  a-,7A,)'«i  •  ■  •  {T)j.  -  a/TV)'"'']  z  =  0.  then 

z  =  *„(;/  +  a^x)  +  -t^^Jy  +  cx^x)  +  •  •  •  +  x'"i  -1*1  ,„^{y  +  a^r)  +  •  .  • 

where  the<f>"s  are  all  arbitrary  functions.  This  gives  the  soiutinn  of  the  reduced  equa- 
tion in  the  sinq)lest  case.    What  terms  would  correspond  tu  (7J,.  —  cxlJy  —  /3)"'2  =  0  ? 

11.  Write  the  solutions  of  the  equations  (or  equations  reduced)  of  Ex.8. 

12.  State  the  rule  of  Ex.  0  (7)  as:  Integrate  7?(.r,  y  —  ax)  with  respect  to  /  and 
in  the  result  change  </  to  //  +  ax.  Ai)})ly  this  to  olitaining  particular  solutions  of 
Ex.  8  (5).  (e).  (rj)  with  the  aid  oi  any  short  cuts  that  are  analogous  to  those  of 
Chap.  VIII. 

13.  Integrate  the  following  eijuations: 

(a)  (If.  -  L':,,  +  1),,  -  1  )  z  =  C( -s  ix  +  -i  //)  +  e",         (/3)  a'2,.2  +  9  xy^  +  yH"-  =  x'^  +  2/^, 

(7)  (If  +  7),,,  +  7a,  -l)z  =  sin  (X  +  2 y),  (5)   ;•  -  ^  -  377  +  3 7  =  e''  +  - ?', 

( e )  (If  -  2  l),^  +  7>2 )  2  =  X  -  -K  (X)  r-t+p-\-?,q-2z^  e--:  -  •-'-  x-y. 

{n)  { If  -  IJ,J),  -  2  7/;  +  2  7>,  +  2  7J,)  2  =  f^^-  +  -='y+  sin  (2  .r  +  y)  +  xy. 

14.  Try  Mongers  method  on  these  e(iuations  of  the  second  older  : 

(a)  q-r  -  2pqH  +  pH  =  0.  (^)  r  -  aH  =  0,  (7)  r  +  ,s  =  -  p. 

(5)   7(1  +  q)r-  (]>  +  q  +  ■2pq).-<+p(l+p)t  =  0,  (e)  x-r  +  2xys  +  yH  =  0, 

(  f )   (/'  +  '■<l)-i-  -  2  ('<  +  <-'y) '"  +  '-7M  'S  +  ("  +  nA-t  =  0,        (7;)  /■  +  hiH  =  2  «.s. 

if  any  simpler  method  is  available,  .state  what  it  is  and  apply  it  also. 


280  DIFFERENTIAL  EQUATIONS 

15.  Show  that  an  equation  of  the  form  Rr  +  Ss  +  Tt  +  U {rt  —  s^)  =  V  neces- 
sarily arises  from  the  elimination  of  the  arbitrary  function  from 

Ui{x,  y,  z,  p,  q)  ^f[u^_{x,  y,  z,  p,  g)]. 

Note  that  only  such  an  equation  can  have  an  intermediary  integral. 

16.  Treat  the  more  general  equation  of  Ex.  15  by  the  methods  of  the  text  and 
thus  show  that  an  intermediary  integral  may  be  sought  by  solving  one  of  the  systems 

IJdy  +  X^  Tdz  +  X^  Udp  =  0,  Vdx  +  \Edy  +  \  Udq  =  0, 

Udx  +  Xj'uly  +  \.,Udq  =  0,  I'dy  +  X.,Tdx  +  KUdji  =  0, 

dz  —  pdx  +  qdy,  dz  —  pdx  +  qdy, 

where  \  and  X^  are  roots  of  the  equation  \-{ilT.+  UV)  +  XL^S  +  U'^  =  0. 

17.  Solve  the  equations  :  (a)  s'^  —  rt  =  0.  (/i)  ,v-  —  rt  —  a-, 

(y)  ar  +  bs  +  ct  -{-  e  {rt  —  s^)  =  h,         (5)  xqr  +  ypt  +  xy  («'-  —  rt)  —  pq. 


PART  III.  IXTEGRAL  CALCULUS 


CHAPTER   XI 


ON   SIMPLE   INTEGRALS 
118.  Integrals  containing  a  parameter.    Consider 

%J  .In 


(1) 


a  definite  integral  which  contains  in  tlic  integrand  a  i)aranieter  a.    If 
tlie  indefinite  integral  is  known,  as  in  the  ease 


/'•'" 


axdx  =  -  sin  ax. 
a 


l'""' 


(ixdx  =  -  sin  ax 
a 


it  is  seen  that  the  indefinite  integral  is  a  function  of  x  and  a,  and  that 
the  definite  integral  is  a  function  of  a  alone  because  the  variable  x 
disappears  on  the  substitution  of  the  limits.  If  the  limits  themselves 
depend  on  a,  as  in  the  (tase 


/' 


(•OS  axdx  =  —  sin  ax 
a 


=  -  (sin  a"^  —  sin  1), 
a  ' 


the  integral  is  still  a  function  of  a. 

In  many  instances  the  indefinite  integral 
in  (1)  cannot  be  found  explicitly  and  it  then 
becomes  necessary  to  discuss  the  conti- 
nuity, differentiation,  and  integration  of  the 
function  (^(a)  defined  by  the  integral  with- 
out having  recourse  to  the  actual  evaluation 
of  the  integral;  in  fact  tliese  discussions 
may  be  required  in  order  to  effect  that 
evaluation.   Let  the  limits  x^  and  x^  be  taken      '' 

as  constants  independent  of  a.  Consider  the  range  of  values  x.^  ^x^x^ 
for  x^  and  let  a^-^(f^a.^  be  the  range  of  values  over  Avhich  the  func- 
tion <^(a)  is  to  be  discussed.  The  function /(a-,  a)  may  be  ])h)tted  as 
the  surface  ,v  =  f(x,  a)   over  the   rectangle  of  values   for   {x,  a).    Tlie 

281 


282  INTEGRAL  CALCULUS 

value  <^  (a,)  of  the  function  when  a  =  n^  is  then  the  area  of  the  section 
of  this  surface  made  by  the  plane  (i  =  ((;.  If  the  surface  f(.r,  a)  is  con- 
tinuous, it  is  tolerably  clear  tliat  the  area  <t>{'i:)  will  be  continuous  in  a. 
The  f miction  ^(^t)  Is  continuous  If  f{j',  a)  Is  continuous  In  the  two  varia- 
bles (x,  a'). 

To  discuss  the  continuity  of  (p{a)  form  the  difference 

<p{a  +  Aa)  -  ct>{a)=  f'^\f(x,  a  +  Aa)  -f{x,  a)]  dx.  (2) 

Now  (f>{a)  will  be  continnous  if  the  difference  0(a  +  An-)  —  (p{a)  can  be  made  as 
small  as  desired  by  taking  Aa  sufficiently  small.  If  f(x,  y)  is  a  continuous  func- 
tion of  (x,  ?/),  it  is  possible  to  take  Ax  and  Ay  so  small  that  the  difference 

\f{x  + Ax,y  +  Ay)-f{x,y)\<e,        |Aj-|<5,         \Ay\<5 

for  all  points  (j,  y)  of  the  region  over  which /(j,  y)  is  continuous  (Ex.  3.  p.  92). 
Hence  in  particular  if /(x,  a)  be  continuous  in  {x,  a)  over  the  rectangle,  it  is  pos- 
sible to  take  Aa  so  small  that 

|/(J,  a+ Aa)-/(x,  a)|<e,         !Aa|<5 

for  all  values  of  x  and  a.    Hence,  by  (Go),  \).  25, 

\4>{a  +  Aa)  -  ct>{a)\=:\  ('''[/ {x.  a  +  Aa)  -f{x.  a)]  */./;|<  f'^'a?,?  =  e{x^  -  x^^). 

It  is  therefore  proved  that  the  function  4>{a)  is  continuous  provided /(.r,  a)  is  ccm- 
tinuous  in  the  two  variables  {x.  a):  for  e(,f^  —  x^)  may  be  made  as  small  as  desired 
if  e  may  be  made  as  small  as  desired. 

As  an  illustration  of  a  case  where  the  condition  for  continuity  is  vi(jlated.  take 

/il      (X(Jx  c  I  ^ 

—^ ~-^  =  tan-1^     =  cot-la     if     a  ?i  0,     and     0(0)  =  0. 
u  a-  +  X-  a  0 

Here  the  integrand  fails  to  be  contintious  for  (0.  0) :  it  becomes  intinite  when 
{x.  a)  =  {().  0)  aloni:  any  cui'vc  that  is  not  tangent  to  a  —  0.  'riie  function  0(a)  is 
detined  for  all  values  of  a  so.  js  eipial  to  cnt^io:  wlien  a  ^  0.  and  slmuld  there- 
fore be  e(iual  to  i-Tr  when  <(  =  0  if  it  is  ti)  be  cnntiiiuous.  whereas  it  is  eipial  tu  0. 
The  importance  nf  the  impnsitioii  of  tlie  condition  that /(,r.  a)  be  continuous  is 
clear.  It  sho\dd  ncjt  be  intVrred.  howt'ver.  that  the  function  0(a)  will  necessarily 
be  discontinuous  when /(,/•.  a)  fails  of  eontinnity.    For  instance 

r'^      dr  1        , ,-  1 

0(a)  ^       —^ — rr-^(\  a  +  l-^  a),         0(0)  =  -. 

^  0  y  a  -[■  X      ^  ^ 

This  function  is  continuous  in  a  for  all  values  asf):  yet  the  intorrrand  is  dis- 
continuous and  indeed  becomes  intinite  at  (0.  Oi.  Tlie  condition  of  continuity 
imposed  on  /(/.  a)  in  the  theorem  is  suffii'lmt  to  insure  tlie  comiiniity  of  0(a) 
bvit  by  no  nirdus  nc-issury :  wiieii  the  condition  is  not  satisfied  sonii'  clo.-er  exami- 
nation of  the  problem  will  sometimes  disclose  the  fact  that  0  ((t)  is  still  continuous. 

In  case  tlie  limits  of  tlic  integral  are  functions  of  a,  as 

(/)  (a)  =    I  /■(.'•,  ^0  (Ir,  a .  g  a  ^  a^,  (3) 


ox   ST:\rPLE   IXTEfJKALS 


28-3 


the   function  <^(t)  will  surely  he  continuous  if  f(p',  a)   is  continuous 
over  the  ret^iou  hounded  hy  the  lines  <x  =  d^^,  ft  =  a^  aiul  the  curves 
X  =  i/o('0'  *^'i  ~  (/ii'^)>  '^^^^^^  ^^  ^^^^  functions  [/^in)  and  f/^('()  are  continuous. 
For  in  this  case 

/> .'/,  (a  +  Art-) 

<p{a  +  Aa)  -  <p  {(x)  =  f{x,  a  +  Aa)  dx 

J" .'/,(")  r"if"^ 

f{.r.  a)dx=  I  f{x,  a+  Aa)dx 

0„(a)  -','/„(a  +  A<0 


I  /■(.f ,  a  +  Aa)  dx 


r  yi(rt) 
•^."oC'O 


fr  +  A(()  —f{x,  a)]dx. 


The  absdhUe  values  may  be  taken  and  the  inte-     A/ 
,2;rals  rechiced  by  ((i.")^,  ((i-V),  p.  25. 

\4>[a  +  Aa)-  <p{a)\<e\<j^{a)-  (/^{a)\  +  \f{^,.  a  +  Aa)\\Af/,\  +  \f{^^,  a  +  Aa')\\Ag^\, 

where  ^^  and  ^^  are  vahies  of  x  between  f/,^  and  r/,j  +  Af/,,.  and  r/^  and  f/,  +  Af/p  By 
taking  Atr  small  enongli.  (/^(a  +  Aa)  —  f/|((r)  and  f/„(a  4-  Aa)  —  f/|,(a)  may  be  made 
as  small  as  desired,  and  hence  A(p  may  be  made  as  small  as  desired. 

119.    To  hud  tl/e  (Icrlrdtlfi'  of  a  fiini-fhni  (f>{'i.)  dejined  hij  an  Intt'jjrul 
confii'inln'j  a  jHiraincfcr,  foi'Ui  the  quotient 

A<^  _  «^  (a  +  \n')  —  cfy  (a) 
Act 


A'( 

X1l^  (.t  +  Alt)  /^.^iCn) 

/'(.'•,  a  +  ^n)d.r  -   /         /(x,  a)dx 
.  „((t  f  Aa)  t7r/„(rt; 


Aa; 


■  dx  + 


+  Aa) 


r-'A  +  ^^f.,.,  r.  +  Aa) 
!  A^f 


f/,r. 


The  transformation  is  made  hv  (''>3),  p.  2.").  A  further  ]'(Hluction  may 
he  made  in  the  last  two  integrals  hy  (05'),  p.  25,  Avhich  is  the  Theorem 
of  the  Mean  for  integrals,  and  the  i.itegrand  of  the  hrst  integral  may  be 
modified  by  the  Theorem  of  the  Mean  for  derivatives  (p.  7,  and  Ex.  14, 
p.  10).    Then 


A^ 

Art- 

and 


.'/„(«) 


f:(x.  a  +  e\a)  dx  -  f(L,  a  +  Aa)  "^f'  +  f{P,  a  +  Aa)  -^ 
J  -■   Aa'  ■  Aa' 

da        I    ,  ^     c<i  '     ■  0      /  ^/^j.       .     -  1      /  ,/,j. 


(4) 


A  critical  examination  of  this  Avo)-k   shows  tliat  the  derivative  (f>'{n) 
exists  and  may  lie  ol.)tained  ly  (4)  in  case /',^  exists  and  is  continuous 


284  INTEdKAL   CALCULUS 

ill  (./•,  a)  and  ^o('«),  r/iC")  '^i'^'  (liftV'i-entiabli".  In  tlic  })articular  case  that 
the  limits  <j^  and  rj^  are  eoiistants,  (4)  reduces  to  Lelbnbi's  Rale 

which  states  that  tlie  derirafire  of  a  function  dejined  hij  an  InfpfjroJ 
tr It li  fixed  limits  moij  he  oldalned  hi/  differentia tlnr/  under  the  sltjn  (f 
Integration.  The  additional  two  terms  in  (4),  when  the  limits  are  varia- 
ble, may  be  considered  as  arising  from  (66),  p.  27,  and  Ex.  11,  p.  30. 

This  process  of  different  la  tlnr/  under  the  sign  of  integration  is  if 
frequent  use  In  evaluating  the  function  <f>('t:)  in  cases  where  the  indeti- 
nite  integral  of /(.r,  a)  cannot  be  found,  but  the  indefinite  integral  of 
f^  can  be  found.    For  if 

«^(^0  =  f  /C'"^  '')'^-'->     then     '■—-=[  \Cdr  =  ^(a). 

Now  an  integration  witli  respect  to  a  ^\\\\  give  (^  as  a  function  of  a 
with  a  constant  of  integration  which  may  be  detenuincd  ly  tlie  usual 
method  of  giving  a  some  special  value.    Thus 


Jo    i"y-^'  '^''    Jo     ^^y-'-         Jo 


Ix. 


Hence  ^  =  — — r  x''+'    =  — — - ,  <^  (a)  =  log  (a  +  1;  +  C. 

da       a  -{-1  |q      ct  +  1  ^  ^  ^  -^ 

But  <^  (0)  =  To  (/,/■  =  0     and     <^  (0)  =  log  1  +  C. 

fVO 

r^  x"  —  1 

Hence  (b(a)=  \ ^/r  =  log  (^r  +  1). 

Jo    1^'y  ■'■ 

In  the  way  of  coniineiit  upcm  this  evaluation  it  may  be  remarked  that  the  func- 
tions (x"^  —  l)/lo,i,^,r  and  x"  are  cnntinuous  functions  of  (,r,  a)  for  all  values  of  x  in 
the  interval  0  ^  .r  ^  1  of  inteuration  and  all  positive  values  of  a  less  than  any 
assijined  value,  that  is,  0^(f^7\'.  The  conditions  which  permit  the  differen- 
tiation under  the  si,i,ni  of  inteuration  are  tlierefore  satislied.  This  is  not  true  for 
neirative  values  of  a.  When  a<0  the  derivative  ./•"  becomes  infinite  at  (0.  0).  The 
method  of  evaluation  cannot  therefore  lie  applied  without  further  examination. 
As  a  matter  of  fact  4>(a)  =  ]o<j:(ir  +  ^)  is  defined  for  a->  — 1,  and  it  would  be 
natural  to  think  that  sduie  method  could  be  found  to  justify  the  above  formal 
evaluation  of  the  integral   \vhen   —  ]  <  re  ^  A'  (see  Chap.  XIII). 

To  illustrate  the  applicatinn  of  the  rule  for  differentiation  when  the  limits  are 
functions  of  a.  let  it  be  i-iMjuireil  to  differentiate 

^a\r<>  —  \   ,  (J(p        /•''      ,         a-"  —  \  a' —  1 

(P{cx)=    I       dx.  -     =  I     x'Hlx -\ a  — . 

«/a      log  x  da      J  a  \o"  a  \o'j.  a 


OK  SIMPLE  INTEGKALS  285 


dip 
da 


_  a^ ^^^  +  1_  1  1^ L^-2<r  _  ^<r  _  (.^  +  1      . 

a +11  J      \ogal  J 


This  formal  result  is  only  good  subject  to  the  conditions  of  continuity.  Clearly  a 
nnist  be  greater  tlian  zero.  This,  however,  is  the  only  restriction.  It  might  seem  at 
llrst  as  though  the  value  x=  1  with  logx  =  0  in  the  denominator  of  (x"  —  l)/logu; 
would  cause  difficulty ;  but  when  x  =  0,  this  fraction  is  of  the  form  0/0  and  has  a 
finite  value  which  pieces  on  continuously  with  the  neighboring  values. 

120.  Tlic  next  problem  would  be  to  find  f/ie  Integrdl  of  a  fiinrflon 
(h\ti)u'<l  III/  (1)1  infcijrdl  coyitdmlnr/  a  i:)arameter.  The  attention  will  be 
restricted  to  the  case  where  tlie  limits  a-^  and  x^  are  constants.  Consider 
the  integrals 

r  <^(a)rAr=   r  .  r   /(^,  a)dx-iJa, 

where  a  may  be  any  point  of  the  interval  <x^  ^  (X  ^  <x^  of  values  over 
which  ^  (a)  is  treated.    Let 

Then     <I>'(a)=  j     '•  ,  -  (     /(.>;  a)da- dx  =   I    \f(.r,a)d.r  =  (t>(a) 

by  (4'),  and  by  (06),  ]>.  27;  and  the  differentiati(^n  is  Icoitimate  if  ,/'(•'', ''') 
be  assumed  continuous  in  (,»•,  a).   Now  integrate  with  i-es})ect  to  a.    Then 

Ja„  Jag 

But  $(a'J=  0.    J  fence,  on  substitution, 

^('^)=  f  '•  f  /(■r,n)dn.d.r=   C  cf,(n)dft=   C  •  C  f(x,n)dx-dn.   (5) 

Hence  appears  the  rule  for  integration,  nanu>ly,  i/ifcf/ri/te  under  flie, 
sir/n  of  'inti'ijraf'iini.  Tlie  rule  has  here  been  obtained  by  a,  trick  from 
the  previous  rule  of  differentiation;  it  could  be  proved  directly  by 
considering  tlie  integral  as  the  limit  of  a  sum. 

It  is  interesting  to  note  the  interpretation  of  this  integration  on  the 
figure,  p.  281.  As  ^('•t)  is  the  area  of  a  section  of  the  surface,  the 
product  ^{ix)da  is  the  infinitesimal  volume  under  the  surface  and 
included  between  two  neighboring  planes.  The  integrid  of  ^{'x)  is 
therefore  the  volume  *  under  the  surface  and  boxed  in  by  the  four 

*  For  the  "  voluiiu^  of  a  solid  with  ])ar;dlcl  bases  and  variable  cross  section''  see 
Ex.  10,  p.  10,  and  §  35  witli  Exs.  20,  12o  thereunder. 


286  l^TEdRAL   CALCULUS 

planes  a  =  (i^,  a  —  a,  .r  =  ,r^,  x  =  ,r^.  Tlie  geometric  significance  of 
the  reversal  of  the  order  of  integrations,  as 

V  =1      ■  I      /(.'',  '()  <lfi  ■  dx  =1       ■   \      /(■'',  n)  dx  ■  da, 

is  in  this  case  merely  that  the  volume  may  be  regarded  as  generated 
by  a  cross  section  moving  parallel  to  the  «rt--plane,  or  l)y  one  moving 
l)arallel  to  the  '-./'-plane,  and  that  the  evaluation  of  the  ^'olume  may 
be  made  by  either  method.  If  the  limits  x^  and  x^  depend  on  a,  the 
integral  of  <^(''i)  cannot  be  found  by  the  sim])le  rule  of  integration 
under  the  sign  of  integration.  It  should  be  remarlved  that  integration 
under  the  sign  may  serve  to  evaluate  functions  defined  b}'  integrals. 

As  an  illustration  of  iiitegi'atioii  under  tlie  sipn  in  a  case  where  the  niethrxl  leads 
to  a  function  whieh  may  be  considered  as  evaluated  h\  tlie  nu'thod,  cdnsider 

f  1      ,  1  r^    ,    .  -,  r^  da        ,6  +  1 

4>{a)^  \    x-dx  =  -  -- .  \    <P  (a) da  =  /    ——  =  log  —- - . 

^0  a  +  1  J  a  Ja  a  ■\-\  a  +  1 

Xh                               p\         rkh                                 p  \     pa      a  ^  h                pi  j^h  j'a 
(p{a)da—  I      •    I    x"da-dx=  \    -^—         dx  -  (     ^ '—dx. 
.;                                Jo        Ja                                  Jo    log  J'   a=it             Jo        log  X 

n\  j-h  yn  ])  A.   \ 

Hence  | dx  =  loi>- =  ^p  (a,  h),         (i  =  0,         /;  ^  Q. 

J  0      log  X  "  a  +  1 

In  this  case  the  integrand  contains  two  parameters  a.  h.  and  the  function  deliiied 
is  a  function  of  the  two.  If  <i  ~  0,  the  function  reiluces  to  one  previously  found. 
It  would  be  possible  to  repeat  the  integration.    Thus 

r    --"^dx  =  log(«  +  ]).  r  loi,r((i-  +  I)  da  =  (a  +  1)  lou' (a  +  \)  -  a. 

Jo     logx  Jo 

da-dx=\ — -^^^r?x  =  (,T+ l)log(a  +  l)-a. 

Jo      Jo      logx  Jo  (logx)- 

This  is  a  new  form.    If  here  a  be  set  eijual  to  any  number,  say  1,  then 


Jo 


^      -—'-(?x  =  2  loir  2-]. 


(logx)^ 

In  this  way  there  has  been  evaluated  a  detiuite  int(\L:i'al  wliich  depends  on  no 
parameter  and  which  miy-ht  liave  been  difticult  to  evaluate  directly.  77/c  introdur- 
lion  iif  (I  pdrdnictcr  dnd  its  .sidisequod  cqiuduni  to  <i  particular  value  i.s  of  frequent  use 
in  evaluating  definite  integrals. 

EXERCISES 

1.  Evaluate  directly  and  discuss  for  contiuuity,  0  ^  a  ^  1: 

fi     <t-'/x  /-I        dx  ^  ^     r^       xdx 

Jo     a- +  xJ  Jo    ^'a-^^.r-2  ^'>    ^  a-' +  .f- 

2.  If /(x,  a,  ji)  is  a.  function  containing  two  jtarametci-s  and  is  coiitiinious  in 
the  three  varialiles  (x.  a.  (S)  when  x,,  ^  x  ^  x,.  a,,  ^  (C  ^  (C,.  [i^,  ^  fi  ^  [i^.  show 

I      /(.'•.  a.  ii)dx  =  f/Wi!.  li)  is  continuous  in  {a.  ^i). 


ON   SliMTLE   INTEGKALS  287 

3.  Differentiate  and  lience  evaluate  and  state  the  valid  range  for  a  : 

(d)     I     lo.u  (1  +  a  ens  x) (Ix  =  tt  log , 

Jo       '  2 

,    ,     r^ ,      ,,        -.  o    ,  fTrlogcr-,  a^  ^^  1 

(/^i)  log  (1  -2  a  cos  X  +  ^1-2)  dx  =  -i  ^      ^,  ^'        -     . 

4.  P'ini'  tlie  derivatives  witliont  previously  integrating: 

J/^^i^~  <r  2  r"'  X  f>  ax   Z-     '1 

-  tan  axdx,         (j3)     (      tan-i  —  dx,         (7)     (       e   «'-    dx. 

5.  Extend  the  assumptions  and  the  work  of  P>x.  2  to  find  the  partial  deriva- 
tives (p'^  and  (p'p  and  the  total  differential  dcp  if  Xq  and  x^  are  constants. 

6.  Prove  the  rule  for  integrating  under  the  sign  of  integration  by  the  direct 
method  of  treating  the  integral  as  the  limit  of  a  sum. 

7.  From  Ex.  6  derive  the  rule  for  differentiating  under  the  sign.    Can  the  com- 
plete rule  including  the  case  of  variable  limits  l)e  obtained  this  way  ';' 

po  (■'■.  f ) 

8.  Note  that  the  integral   |  /(x,  a)  (Zx  will  be  a  function  of  (x,  a).    Derive 

"  ■'o 
formulas  for  the  partial  derivatives  with  respect  to  x  and  rr. 

9.  Differentiate  :  (rr)  -'^  -   f' sin  (x  +  a)  dx,     (/3)    '-    f    '"x^cZx. 

f  (t  «^o  dx  Jo 

10.  Integrate  under  the  sign  and  hence  evaluate  by  subse(jueut  differentiation  : 

{a)     I    x'Mogxdx,         (/3)     |   'xsinaxdx,         (7)     (    xsec^axdx. 

Jo  ^'  Jo  Jo 

11.  Integrate  or  differentiate  lioth  sides  of  these  equations  : 

x"(?x  =-- to  show       I     x"  (log  x)"dx  =  (—  1)" , 

0  <i  +  1  -^0  '  ^  («+])"  +  !' 

r  ^       dx  TT  ,  /-  -  dx  TT  1  .  .S  •  .5  .  .  .  (2  H  -  1) 

(/3)     I       = =     to  show       I       — = ^, , 

Jo     X- +  a       2V a  '''>     {x-'"  +  a)"  +  >■       2  2  •  4  •  0- •  •  2  n  ■  a"  + ^ 

(7)  c-"'^  COS  mxdx  = to  show    | dx  =  -  log    — : , 

Jo  a- +  ?/t-  Jq         xsecv/ix  2     '    \a- +  in-J 

c-"''sln  ?«xdx  = to  show    | dx  =  tan-i tan-i  — , 

0  li- -\- 1)1-  Jo        xcsc?«x  )n  m 

(e)     ( =  -    -_^^ .   to  tnid    / ,     f     log , 

Jo      a'—  COSX        ■^^/(^l■^_l  Jo      ((C—  cosx)"^      Jo  «—  COSX 

•^  X"  -i(/x  TT  „    ,     /"=°  x«  -1  Ion'  xdx       r  '^  /''  - 1  —  X"  -1 


dx. 


r-"  x'^-'-dx  TT  „    ,     /"=°  x'^ -Moi;',rax       r 

(i-)     /  = to  find    (       ^^^ ,     I 

Jo       1  +  X        sin  -rra  Jo  1  +  x  Jo      (1  +  x)  logx 

Kote  that  in  (/3)-(5)  the  integrals  extend  to  infinity  and  that,  as  the  rules  of 
the  text  have  been  proved  on  the  hypothe.sis  that  the  interval  of  integration  is 
finite,  a  further  justification  for  applying  the  rules  is  necessary  ;  this  will  be 
treated  in  Cliap.  XIII,  but  at  this  point  the  rules  may  be  applied  formally 
without   justification. 


288  INTEGRAL  CALCULUS 

12,   Evaluate  Iw  any  means  these  inteijrals  : 

^    '   Jo  a  \](;       4/ 

,     z'  o  li  i^M  1  +  (■<  IS  a  r<  >s  ,r )  ,         1  ,  tt-  A 

(/3      f  -  ^^^^ UU  =  -  aM, 

Jo  co.s.f  2  \  4  / 

7r 

(7)     I    "  lo.LT  (a-cos-x  +  /3'-sin'-.r)  f7j  =  Trlog — -^ — -, 


(5)     (" 


r(.-a.r  (■US  jixflr 


a  +  /(sin  X    (Zx 


(e)     I    "  loo; — 

Jo        "a  —  6sinx  sinx 


b  <a, 


dx. 


r  ^  Inix  (1  +  k  cosx)   ,  .       ,  , 

(f)       I  '     ' '(ZX  =  TTsill-'A-, 

Jo  CdSX 

(ff)     f     lncr/(rt  +  x)r7x  =    (  loir /-(x)  r?x  =     I      ],,-■  --^L^rZr/  +   I      log/{x) 

Jo  Jii  J'J  ,/  (")  Jo 

121.  Curvilinear  or  line  integrals.    It  is  i'aiuiliar  that 


is  tlie  area  l)et\v('en  the  curve  //=/'(.;■).  tlic  ./■-axis,  and  tlie  ordinates 
.;■  =  11^  X  =  //.  The  fonnitla  may  l)e  used  to  evaltiate  more  eom])lieate(l 
areas.  For  instanee,  the  area  Ijetweeu  tlie  juirabola  >/'-  =  ,/•  and  the  semi- 
eubical  parabola  //-  =  ,r^  is 


l/o  i/o  rJ)  sJ() 


'>/.,: 


where  in  the  si^-ond  expression  the  sul)seri})ts  /*  and  N  denote  that  the 
intej^'rals  are  evaluated  for  the  parabola  and  semieubieal  parabola.  As 
a  change  in  the  oi-der  of  the  limits  changes  the  sign  of 
the  inteyi-al.  the  area  mav  be  written 


XI  /^  0  /^  0  /-» 1 


and  is  the  area  br)unded  by  the  closed  curve  formed 
of  the  portions  of  the  ])araliola  and  semieubieal  parabola  from  0  to  1. 
In  considei'ing  the  area  liounded  l)y  a  closed  curve  it  is  convenient  to 
arrange  the  limits  of  the  different  integrals  so  that  they  follow  the  curve 
in  a  definite  ordei-.  Thus  if  one  advances  along  P  from  0  to  1  and  re- 
turns along  N  Irom  1  to  0,  the  entire  closed  curve  has  been  described 
in  a  uniform  dii'ection  and  the  inclosed  area  has  T)een  constantly  on  tin; 
riirht-hand  side  ;    whereas  if  one  advanced  along   .S  from   0  to  1   and 


ON  SIIVIPLE  INTEGRALS 


289 


returned  from  1  to  0  along  P,  the  curve  would  have  been  described 
in  the  opposite  direction  and  the  area  Avould  have  been  constantly 
on  the  left-hand  side.  Similar  considerations  apply  to  more  general 
closed  curves  and  lead  to  the  definition :  If  a  closed  curve  which 
nowhere  crosses  itself  is  described  in  such  a  direction  as  to  keep  the 
inclosed  area  always  upon  the  left,  the  area  is  considered  as  positive ; 
Avhereas  if  the  description  Avere  such  as  to  leave  the  area  on  the  right, 
it  would  be  taken  as  negative.  It  is  clear  that  to  a  person  standing  in  the 
inclosure  and  Avatching  the  description  of  the  boundary,  the  descrip- 
tion would  appear  counterclockwise  or  positive  in  the  first  case  (§  76). 
In  the  case  above,  the  area  when  positive  is 


LsJa  rJx  J  Jo 


,'l.r^ 


(6) 


where  in  the  last    integral  the  symbol  O  denotes  that  the  integral  is  to 
be  evaluated  ai'ound  the  closed  curve  by  describing  the 
curve  in  the  })()sitive  direction.    That  tlie  foi-mula  holds 
for  the  ordinary  case  of  area   under   a  curve   may   l)e 

verified  at  once.    Here  the  circuit  consists  of  the  con-  ^ 

tour  ABB' A' A.    Then  o\   A  BX 

C ;,,Lr  =    r    ydx  +    r      ,jd.r  +    f     f/>/.r  +    f    ydx. 
Jo  J  A  J  n  Jr.'  J  A' 

The  first  integral  vanishes  because  //  =  0,  the  se(,'ond  and  fourth  vanish 
because  x  is  constant  and  (/./■  =  0.    Hence 


that 

Jo 


-£"''"-f?"'- 


It  is  readily  seen  that  the  two  new  formulas 
'y     and     ,1 


Jo 


ydx) 


(') 


'o  «^o 

also  give  the  area  of  the  closed  curve.  The  first  is  proved  as  ((V)  was 
])rove(l  and  the  second  arises  from  tlie  addition  of  the  two.  Any  one 
of  the  three  luay  be  used  to  compute  tlie  area  of  the  closed  curve:  the 
last  has  the  advantage  of  symmetry  and  is  particularly  useful  in  finding 
the  area  of  a  sectoi'.  l>ecause  along  the  lines  issuing  from  the  origin 
y  :  X  =  d y  :  dx  and  xify  —  ydx  =  0  :  the  previous  form  with  the  integrand 
xdy  is  advantageous  when  part  of  the  contour  consists  of  lines  parallel 
to  the  .''-axis  so  that  r/y  =  0 ;  the  first  form  has  similar  advantages 
wlieii  ]iarts  of  the  contour  are  parallel  to  the  //-axis. 


s 


290  INTEGRAL  CALCULUS 

The  connection  of  the  third  formula  with  the  vector  expression  for 
the  area  is  noteworthy.    For  (p.  175) 

dA  =  I  Txdt,  A  =  I    I  Txdr, 

Jo 
and  if  r  =  xi  +  y],      dr  =  idx  +  yJij, 

then  A=   /  rxr/r  =  J  k   /  {xdij  —  ydx). 

Jo  Jo 

The  unit  vector  k  merely  calls  attention  to  the  fact  that  tlie  area  lies 
in  the  .-r//-plane  perpendicular  to  the  r^-axis  and  is  described  so  as  to 
appear  positive. 

These  formulas  for  the  area  as  a  curvilinear  integral  taken  around 
the  boundary  have  been  derived  from  a  simple  figure  whose  contour 
was  cut  in  only  two  points  by  a  line  parallel  to  the  axes.  The  exten- 
sion to  more  complicated  contours  is  easy.  In  the  tirst  })lace  note  that 
if  two  closed  areas  are  contiguous  over  a  part  of  their  contours,  the  inte- 
gral around  the  total  area  following  both  contours,  but  omitting  the  pai't 
in  common,  is  equal  to  the  sum  of  the  integrals.    For 

J  I'Rsr     J  j'QRr     J  I'll     J  i:sr     Jpqk     Jnr     J    qrs/'  pf- 

since  the  first  and  last  integrals  of  the  four  are  in  ()i)po- 

"^         .  Q 

site  directions  along  the  same  line  and  must  cancel.  JJut 
the  total  area  is  also  the  sum  of  the  individiial  areas  and  hence  the 
integral  around  the  contour  /'Q/iSP  must  be  the  total  area.  The  for- 
mulas for  determining  the  ai'ca  of  a  closed  curve  are  therefore  applicable 
to  such  areas  as  may  ha  composed  of  a  finite  number  of  areas  each 
bounded  by  an  oval  curve. 

If  the  contour  liouiuling  an  area  be  expressed  in  parametric  form  as  x=f{t), 
y  —  <p  {t),  tlie  area  may  be  evaluated  as 

ff{t)i>'{t)  dt  =  -  f<p{t)r{i)  at  =  ifif{t)^\t)  -  0  (o/'(0]  dt,  (70 

where  the  limits  for  t  are  the  value  of  t  corresponding  to  any  point  of  the  contour 
and  the  value  of  t  correspondin.i,^  to  the  same  point  after  the  ciu'vo  has  been 
described  once  in  the  positive  direction.    Thus  in  the  case  of  the  strophoid 

fl  —  X 

7/2  =  X- ,     the  line     ?/  —  tx 

a  +  X 

cuts  the  curve  in  the  doul)l(>  point  at  the  orii^in  and  in  oidy  one  other  point;  tlie 
coordinates  of  a  point  on  the  curve  may  be  exx^ressed  as  rational  functions 

J-  =  a  (1  -  r^)/{]  +  r'),  y  =  at  (1  -  r^)/(1  +  t^) 

of  t  by  solving-  the  stroi)hoid  with  the  line  :  and  when  I  varies  from  —  1  to  +  1  the 
point  {x,  y)  describes  the  loop  of  the  strophoid  and  the  limits  for  t  are  —  1  and  +  1. 


ox  SIMPLE  INTEGRALS 


291 


122.   Consider  next  the  meaning  and  tlie  evalnation  of 

f    '[Pi-^;i/)^^-'-  +  Q(->;!/)'^!/l     where     v/ =/(./•)• 


(8) 


This  is  called  a  cuvvlUneat'  or  Line  integral  along  the  cur  re  C  or  y  =  f(^,r^ 
from  the  point  (a,  It)  to  (x,  y).  It  is  possible  to  eliminate  y  by  the  rela- 
tion y  =f{.c)  and  write 


U  a 


[P(,r,/(.r))  +  Q(.r,/(.r))/'(.r)].?.r. 


(9) 


The  integral  then  becomes  an  ordinary  integral  in  .r  alone.  If  the  cyrve 
had  been  given  in  the  form  x  =f(^y')^  it  would  have  been  better  to  con- 
vert the  line  integral  into  an  integral  in  //  alone.  TJie  iiwthod  of  cvalaaf- 
UKj  ihe  Iniegral  is  therefore  defined.  The  differential  of  the  integral 
may  be  written  as 


d 


r\pd.: 

Ja,h 


-c  -f-  Qdy)  =  Pdx  +  Qdy, 


(10) 


Avhere  either  x  and  dx  or  y  and  dy  may  l)e  eliminated  by  means  of  the 
ecpiation  of  the  curve  ( '.    For  further  particulars  see  §  123. 

To  get  at  tlie  meaning  of  the  line  Integral^  it  is  necessary  to  con- 
sider it  as  the  limit  of  a  sum  (compare  §  16).  Suppose  that  the  curve 
C  between  («,  V)  and  (.r,  y)  be  divided  into  n  parts,  that  A./\  and  A//,- 
are  the  increments  corresponding  to  the  /th  part,  and  that  (^,,  ly,)  is 
any  point  in  that  })art.     l'\)rm  the  sum 


=  X  t^'  ^^^"  '?'•)  ^"■'  +  ^^  ^^'■'  ^'■)  ^'1^ 


If,  Avhen  n  becomes  infinite  so  that  A,r  and  A//  eacli 
ajjproaches  0  as  a  limit,  the  sum  o-  ap])roaclies  a 
definite  limit  inde})endent  of  how  Ww.  individual 
increments  A,/',  and  A//,-  approach  0,  and  of  how  the 
point  (^,,  ■>;■)  is  chosen  in  its  segment  of  the  curve, 
then  this  limit  is  defined  as  the  line  iuteural 


lim  o-  =      r      [P  (,r,  //)  dx  +  a  (,/■,  //)  dyl. 
rJa,  h 


(12) 


It  should  l)e  noted  that,  as  in  the  case  of  the  line  integral  which  gives 
the  area,  any  line  integral  which  is  to  be  evaluated  along  two  curves 
Avhicli  liave  in  common  a  portion  described  in  o])posite  directioiis  may 
be  i'e[)laced  by  the  integral  along  so  mucli  of  the  cui'ves  as  not  re})eated  ; 
for  the  elements  of  a  corresponding  to  the  common  portion  are  equal 
and  opposite. 


292 


INTEGRAL   CALCULUS 


That  a  does  approach  a  limit  provided  P  and  Q  are  continuous  functions  of  (x,  y) 
and  provided  the  curve  C  is  monotonic,  that  is,  that  neither  Ax  nor  A*/  changes  its 
sign,  is  easy  to  prove.    I'or  tlie  exi)ression  for  a  may  be  written 

by  using  the  equation  y  =f{x)  or  x  =/~i  (y)  of  ('.    Now  as 

J  "/'  (X,  /(x))  tZx     and     J  "q  (/-i  (//).  y)  (Zy 

are  both  existent  ordinaiy  definite  integrals  in  view  of  the  assumptions  as  to  con- 
timiity,  the  sum  a  must  approacii  their  sum  as  a  limit.  It  may  be  noted  that  this 
proof  does  not  reijuire  the  continuity  oi'  existence  of /'(x)  as  does  the  fornuda  (U). 
In  practice  the  added  generality  is  of  little  use.  The  restriction  to  a  monotonic 
curve  may  be  replaced  l)v  the  assumption  of  a  curve  (J  which  can  be  regarded  as 
made  up  of  a  finite  inunber  of  monotoinc  parts  including  perhaps  some  portions  of 
lines  parallel  to  the  axes.  More  general  varieties  of  (J  are  admissible,  but  are  not 
very  useful  in  practice  (§  127). 

FurtluM-  to  exuiuinc  the  line  iiitogval  and  ai)preciate  its  utility  for 
matliematies  and  })li_vsics  consider  some,  c'xani])les.    Let 

be  a  complex  function  (§  T'.V).    Then 


{Xdc  —  Y(h/)  +  /     I         (Y(/.r  +  X<f;/). 


{!?,) 


It  is  ap])arpnt  that  fln'  'infiijriil  of  ilw  (■oiiipJc.rfKncfinn  !s  flw  sum  of  tico 
line  iiifrf/rcls  In  flu'  ciniijiLcj'  phinc  The  value  of  the  integral  can  be 
comjmted  only  by  the  assuni])tion  of  sonn'  definite  path  ('  of  integra- 
tion and  will  differ  for  different  ])aths  (but  see  §  124). 

\\\  dehnition  iJtc  irarl;  (hmc  hij  a  cdnstdtif  furcr  /•"acting  on  a  particle, 
"which  moNcs  a  distance  .s'  along  a  straight  line  inclined  at  an  an.gle  6  to 
the  force,  is  11'=  /•'.>>•  cos  ^.  If  the  jiath  wei'e  curvilinear  and  the  force 
wei'e  variable.  fJic  d'lfft'rcni'Kil  of  irorl:  woidd  be  talven 
as  (1  ]V  =  l-\-{)<,  9ils,  where  ds  is  the  infinitesimal  arc; 
ami  d  is  the  angle  between  tlie  are  and  the  force. 
Heiu'(^ 


W  =     i  d]V  =    I  FcoS^r/.v  =    I 


F.r/r, 


A' 


wliere   the   ])ath   must  be   known    to    evaluate   the    integi'al  and   where 
the  last  t'xpressiou  is  merely   the  e(|ui\alent  of  tin;  otliers  when   the 


ox   SIMPLE   IXTEGKALS  293 

notations  of  vectors  are  used  (p.  1G4).  These  ex})ressious  may  be  con- 
verted into  the  ordinary  form  of  the  line  integral.    For 

F  =  A'i  +  }'j,         dx  =  hix  +  yf//,         F.r/r  =  X'U  +  l>/y, 

Fros  Oils  =    /        {Xd.r  +  Yd;/), 

h  Ja.  ii 

"where  X  and  ]'  are  the  com])onenrs  of  the  force  alonj^'  the  axes.  It  is 
readily  seen  that  any  line  inti-yral  may  ]»e  given  this  same  inter- 
]>retation.    If 

1=1      Pdj-  +  Qdii.     foi'iu     F  =  7^i  +  Qy 
J 'I,  h 

Xj.  !l  r>  J.  1/ 

Pdx  +  (ldii  =    I        Frv^eds. 

To  the  principles  of  inoineiituin  and  ninnient  of  nKinientuni  (S  80)  may  now  be 
added  the  princii>le  of  work  and  eneriiv  fur  nieelianics.    Consider 

in  '^'^  =  F     and     in  -'-''.  -7r  =  F-<h  =  <l  IT. 
dt-  di- 

^,                                    d  ildx  dx\       1  r."-r   dx       \  dx  d-X       d-x  dx 
Then  —    -      •        =     h     =-     .^, 

dt\ldl   dt!       --^ 'U-'   dt       2, It   dt-       dt-   dt 

or  (Z/Ji--)  =  '''^</r     and     d  C'^^miA  =  dW. 

1,1,        /<r 
Hence  -mv-  —  inv-  =    |    F.'7r  =  II  . 

In  words  :  The  change  of  the  kinetic  cneyy;/  ]  mv-  of  a  pctrticle  moving  tinder  the 
(iction  of  the  resultunt  force  F  is  eijuiil  to  the  u-urk  done  lit/  the  force,  that  is,  to  the  line 
integ-ral  of  the  fon^e  alonii'  the  path.  If  there  were  several  mutually  interactiiii,' 
particles  in  motion,  the  residts  f(.ir  the  energy  and  work  would  merely  tie  added  as 
•2  i  niv-  —  -  1  »H',-J  1^  2  ir.  and  the  tntal  chanue  in  kinetic  eneruy  is  the  total  work 
done  V)y  all  tho  fc.irces.  The  result  irains  its  significance  chiefly  by  the  consideration 
(jf  what  forces  may  be  disrei:arde(l  in  evaluating;  tlie  work.  As  dW  =  F«(/r,  the 
work  done  will  be  zero  if  dx  is  zero  ov  if  F  iuid  dx  are  yjerpendicular.  Hence  in 
evaluatinir  ir,  forces  whose  point  of  application  does  not  move  may  be  omitted 
(for  example,  forces  of  su])port  at  jiivots).  and  so  may  forces  whose  point  of  appli- 
cation moves  normal  to  the  force  (for  example,  the  normal  reactions  of  smooth  curves 
or  surfaces).  When  more  than  one  particle  is  concerned,  the  work  done  by  the 
nuitual  actions  and  reactions  may  be  evaluated  as  follows.  Let  r^ .  r.,  be  the  vectors 
to  the  particles  and  x^  —  r.,  the  vector  joiniuir  them.  The  forces  of  action  and  re- 
action may  be  written  as  ±  c  (ij  —  r.,).  as  they  are  equal  and  opposite  and  in  the  line 
joining  the  pai'ticles.    Hence 

d]r=d]\\  +  d\V.,  =  c  (ti  -  x.,).dx^  -c{x^-  r.,).(Zr, 

=  c  (r^  -  r.).'?  (rj  -  r,)  =  ]  rd  [(r,  -  r._,).(ri  -  r.)]  =  l  cdrj.,, 

where  ;•,.,  is  the  ilistani;e  between  the  iiartii'les.  Now  (7  IF  vanislies  whe.i  and  only 
when  dr^^  vanishes,  that  is,  when  and  only  when  the  distance  between  tl'j  i>ariicles 


294  INTEGRAL  CALCULUS 

remains  constant.  Hence  v:hcn  a  system  of  jjurtkks  is  in  motion  the  change  in  the 
total  kinetic  energy  in  passing  from  one  position  to  another  is  equal  to  the  icork  done  by 
the  forces,  where,  in  evaluating  the  work,  forces  acting  at  fixed  points  or  normal  to  the 
line  of  motion  of  their  points  of  application,  and  forces  due  to  actions  and  reactions  of 
particles  rigidly  connected,  inay  he  disregarded. 

Another  important  application  is  in  tlie  tlieory  of  tliermodynaniics.  If  I',  j).  v 
are  the  ener<ry.  pressure,  volume  of  a  gas  inclosed  in  any  receptacU'.  and  if  dU  and 
dv  are  the  increments  of  energy  and  volume  wlien  the  amount  (///  (if  heat  is  added 
to  the  gas.  then  „ 

dll  =  dU  +  pdv,     and  hence    J/ =       dl' +  pdc 

is  the  total  amount  of  heat  added.   By  taking  p  and  r  as  tlie  indrjicndent  variables, 

The  amount  of  heat  absorbed  by  the  system  will  therefore  not  depend  merely  on 
the  initial  and  final  values  of  {p,  v)  but  on  the  sequence  of  these  values  between 
those  two  points,  tliat  is.  upon  the  path  of  integration  in  the  yjr-plane. 

123.  Let  there  1)6  given  a  simply  connected  region  (p.  89j  lionnded  by 
a  closed  curve  of  the  type  allowed  for  line  integrals,  and  let  P(.r,  >/)  and 
Q(.i\  II)  be  continuous  functions  of  (.>•,  //)  over  this  region.  Then  if  the 
line  integrals  from  (a ,  li)  to  (.r,  y)  along  tw(j  paths 

r  '  Pdx  +  Qdi,  =      r     ]'<l.r  +  Q'hj 

cJa,b  vJti.h 

are  ec^ual,  the  line  integral  taken  around  the  combined  path 

r"'+  r  =  f  p<Lr+ <i,i,,=:{) 

cJa.h  rJ.r.ij  J^ 

vanishes.  This  is  a  cort)llary  of  the  i'art  that  if  the  order  of  description 
of  a  ctirve  is  reversed,  the  signs  of  A.'-,  and  A//,-  and  liciice  of  the  line 
integral  are  also  reversed.  Also,  conversely,  if  tlie  in- 
tegral around  tlu;  chxsed  circuit  is  zero,  the  iiitt^grals 
from  any  point  {a,  h)  of  the  circuit  to  any  other  point 
(.r,  //)  are  equal  when  evaluated  along  the  two  different 
parts  of  the  circuit  leading  from  (/'.  //)  to  (./•.  //). 

The  chief  value  of  these  ol)servations  ai'iscs  in  their  application  to 
the  case  where  1*  and  Q  hap])en  to  be  such  functions  that  tlie  line  inte- 
gral aromid  any  and  evci-y  closed  ])atli  lying  in  the  iTgion  is  zero.  In 
this  case  if  (",  /')  be  a  fixed  jioint  and  (./•.  //)  lie  any  jioint  of  the  region. 
the  line  integral  from  ('/.  I>)  to  (./•.  //)  along  any  two  jiaths  lying  within 
the  region  will  be  the  same:  for  the  two  ])aths  may  lie  considered  as 
forming  one  closed  jiatli,  and  the  integral  aromid  tliat  is  zero  by  hy- 
pothesis.   The  value  of  the  integral  will  thertd'ore  not  depend  at  all  on 


ox  8LMPLI<:   INTEGRALS  295 

the  patli  of  integration  but  only  on  the  final  point  (x,  ij)  to  which  the 
inte<rration  is  extended.    Hence  the  integral 


x: 


[P  (./■,  //;  dx  +  a  (,r,  II)  (hj-\  =  F (x,  y),  (14) 


extended  from  a  fixed  loAver  limit  {a,  h)  to  a  variable  upper  limit  {x,  y), 
must  l)e  a  function  of  (,/■.  y). 

This  result  may  be  stated  as  the  tlieorem  :   Tlie  necessary  and  suffi- 
cient condition  tliat  the  line  Integral 


L 


[Pix,y)dx  +  Qix,y)dy^ 

'  II.  h 

define  a  single  valued  function  of  (x,  y)  orer  a  simply  connected  region 
Is  that  tlie  circuit  integral  toh-cn  oround  any  and  ecery  closed  curce  In 
the  region  shall  he  zero.  This  theorem,  and  in  fact  all  the  theorems  on 
line  integrals,  may  l)e  innnediately  extended  to  the  case  of  line  integrals 
in  space, 


x: 


IP  (x,  y,  -:)  dx  +  Q  (x,  y,  z)  dy  +  R  (x,  y,  z)  dz^  (15) 


If  tlie  Integral  ahout  erery  closed  patJi  Is  zero  so  that  the  Integral  from 
a  fixed  lower  limit  to  a  varlahle  upper  limit 


F(:'',y)=  f     PQ';y)dx-\-mx,i 

J  a,  h 


l/)(^!/ 


defines  a  function  F[x,  //),  that  function  lias  continuous  first  pjartlal 
derlvatlccs   and  hence   a    total  dlfferentud,   namely, 

—  =  P,  —  =  Q,  dF  =  Pdx  +  Qdu.  (16) 

ex  cy  ^  '     -^ 

To  prove  this  statement  a])])ly  the  definition  of  a  derivative. 

l\lx  +  Qdy  -  j        ]\lx  +  (Idy 

Xow  as  the  integral  is  independent  of  the  path,  the  integral  to 
(,/•  +  A.r,  //)  may  follow  the  same  i)ath  as  that  to  (x,  y),  except  for 
the  passage  from  (./•,  //)  to  (./■  +  A.r,  //)  which  may  be  taken  along  the 
straight  line  joining  them.    Then  A//  =  0  and 


cF       ,.      AF 

-;: —  =    lim    

=  lim 

CX         A.,-  =  oA.r 

A.'-  =  0 

A.7 
A 


^  =  \     f     ^"''Pf-r,  y)dx  =  ^P($,  y)Ax  =  P(i,  y), 


296  I^TEGKAL  CALCULUS 

by  the  Theorem  of  the  Mean  of  (65'),  p.  25.  Now  when  A./-  =  0,  the 
value  ^  intermediate  between  x  and  x  +  A,/.-  will  approach  ./•  and  P  (^,  ij) 
will  approach  the  limit  P{x,  ij)  by  virtue  of  its  continuity.  Hence 
AF/Aa;  approaches  a  limit  and  that  limit  is  /-"(.>•,  y)  =  cF/cx.  The  other 
derivative  is  treated  in  the  same  way. 

Jf  the  integnnid  Pdx  +  Qdy  of  a  line  integral  is  the  total  diffi^rential 
dF  of  a  single  valued  function  F{x,  y),  then  the  integral  ahoat  any  closed 
eireiiit  is  zero  and 

f  ''pdx  +  Qdy  =  f  '\lF=  F(x,  y)  -  Fia,  h).  (17) 

If  equation  (17)  liolds,  it  is  clear  that  the  integral  around  a  closed  path 
will  be  zero  provided  F{x,  y)  is  single  valued:  for  F[x.  y)  must  come 
back  to  the  value  F{a,  h)  when  (x,  y)  returns  to  (a,  h).  If  the  function 
were  not  single  valued,  the  conclusion  might  not  hold. 

To  prove  the  relation  (17),  note  that  by  definition 

jdF  =  jrdx  +  Qdy  =  liin^  [P(t,-.  ,;,)A/,-  +  Q(t,-.  t,,)^^,] 

and  AF,-  =  P  (t;,  m)  A.f ,-  +  Q  (t,-,  t;,)  Ay,-  +  e^XXt  +  e.,A?/,-, 

where  ej  and  e.,  are  (jnantities  wliich  by  the  assumptions  of  continuity  for  P  and  Q 
may  be  made  uniformly  (§25)  less  than  e  for  all  points  of  the  curve  provided  A/,- 
and  A//,-  are  taken  small  enough.    Then 


2^  { P,  Ar  ,•  +  Qi\y,  )-^M\\<e^(\\jCi\  +  \  Ay. ' ) ; 
=  F{x.  y)  —  F{(i.  h).   the  sum   Zl'iAXi  +  Q,Ayi  appn 

lim^^  [p,A/,  +  Q,A//,]  =  f'''^''i-<-  +  Wy  =  -f'(-^,  y)  -  ^(«-  ^^ 


and  since  2AF,-  =  F{x.  y)  —  F[(t.  h),  tlie  sum  S/',A.f,-  +  Q,Ayi  approaches  a  limit, 
and  that  linut  is 


EXERCISES 

1.  Find  the  area  of  the  loop  of  the  strophdid  as  indicated  above. 

2.  Find,  from  (6),  (7),  the  three  expressions  for  the  integrand  of  the  line  inte- 
grals which  give  the  area  of  a  closed  curve  in  polar  coordinates. 

3.  Given  the  equation  of  tlie  ellipse  x  =  a  cos  ^  y  =  6  sin  t.  Find  the  total  area. 
the  area  (jf  a  segment  from  the  end  of  the  major  axis  to  a  line  jiaraUel  to  the  minor 
axis  and  cuttim:-  the  ellii:>se  at  a  jjoint  whose  i)arameter  is  t.  also  tlie  area  of  a  sector. 

4.  Find  the  area  of  a  segment  and  of  a  sector  for  the  hyperbola  in  its  parametric 
form  I  =  (I  cosh  /.  //  =  h  sinhi. 

5.  Express  the  folium  j-°  +  y^'  =  3axy  in  parametric  form  and  find  the  area  of 
the  loop. 

6.  What  area  is  given  by  the  curvilinear  integral  around  the  perimeter  of  the 
closed  curve  ;•  =  i;  siu''*  J  0 '.'  What  in  the  case  of  the  lemniscate  /•-  =  «■- cos  2  <^ 
described  as  in  makini,'-  the  fiyure  8  or  the  sign  x  V 


ox  SIMPLE  INTEGRALS  297 

7.  Write  for  y  the  analogous  fonn  to  (9)  for  x.    Show  that  in  curvilinear 
coordinates  x  =■  <p{u,  v),  y  =■  ^{u,  v)  the  area  is 


8.  Compute  these  line  integrals  along  the  paths  assigned: 
>i,  1 


=  X     or     y'  =  X-, 
1, 1 


or     y  =  X     or     y^  —  x-, 


(a)     f     x-ydx  +  y"dy,         y-  =  x     or     y 
(/3)     r  '  (x2  +  y)dx  +  (X  +  y-)dy,         7j-  =  x 

«^0.  0 

J"  '•  1  1/ 
~dx  +  dy,         ?/  =  logx     or    y  =  ()     and     x  =  e, 
1,  0    x" 
J-*  J  ■  1/ 
X  sin //tZx  +  2/ cos  xcZ?/,         y  =  mx     or    x  =  0     and     y  =  y, 
0,  0 

(e)  /         {x—iy)dz,     y  =  x     or    x  =  0     and     y  =  l     or     ?/ =  0     and     x  =  1, 

(f)  I         (•'■"  —  (1  +  O-^'y  +  y")'^~i         <iuadrant  or  straight  line. 

9.  Show  that  C Pdx  +  Qdy  =  jVl'-  +  (/-  eos6'(i.s  by  working  directly  with  the 

figure  and  without  tlie  use  of  vectors. 

10.  Show  that  if  any  circuit  is  divided  into  a  number  of  circuits  by  drawing 
lines  within  it,  as  in  a  figure  on  p.  91,  the  line  integral  around  the  original  circuit  is 
etjual  to  the  sum  of  the  integrals  anmnd  the  subcircuits  taken  in  the  proper  (U'der. 


11.  Explain  the  method  of  evaluating  a  line  integral  in  space  and  evaluate  : 

/ni,  1. 1 
(a)     i  xdx  +  '2  ydy  +  zdz,         y-  =  x,         z-  =  x     or     y  =z  z  =  x, 

^0,  0.  0 

y  logxJx  +  y'-dy  +  -dz,      ?/  =  x  —  1.      z  =  x-  or   y  —  logx,      z  =  x. 
1.  0,  1  '  ~ 

12.  Show  that  C I'dx  +  Qdy  +  lldz  ^   ^^  1'-  +lj^+  H-  eos^J.s. 

13.  A  liead  nf  mass  //(  strung  on  a  f rictionless  wire  of  any  shape  falls  frr)m  one 
]>oint  (x,,.  ;/,,.  z„)  tn  the  puint  (Xj.  y^.  z^)  nn  the  wire  under  the  influence  of  gravity. 
SIkiw  tliat  »/;/(2,|  — 2,)  is  tlie  work  dniic  liy  all  the  forces,  namely,  gravity  and 
the  normal  reaction  nf  the  wire. 

14.  If  X  =f{t).  y  =  (j{t).  and/'(/).  (/'((}  be  assumed  continuous,  show 

f  ■'■'  "p  (X,  y)  df  +  Q  (X.  y)  dy  =  f'(p'^+Q  '^^  dt, 
'Jii.b  'Ji„  \     dt  dt/ 

where /(^ij)  =  <(  and  </  {t,^)  =  h.  Nnte  that  this  proves  the  .statement  made  on  page  200 
in  regard  to  the  possibility  of  sul)slituting  in  a  line  integral.  The  theorem  is  also 
needed  for  Exs.  1-8. 

15.  Extend  to  line  integrals  (!"))  in  s]iace  tlie  results  of  §  12o. 

16.  AiKjlc  (IS  II  liuf  intfyriil.  Show  geometrically  for  a  iilane  curve  tliat 
d<p  =  cos(/',  n)d>:/r,  where  r  is  the  radius  vector  of  a  curve  and  ds  the  element  of 


298  INTEGRAL  CALCULUS 

arc  and  (r,  ?j)  the  angle  between  the  nulins  produeeil  and  the  luirnud  to  the  curve, 
is  the  angle  subtended  at  r  —  0  by  the  element  d.s.    Ilenee  show  that 

(P  =  I   ^-^—^  ds=  I ds  =  I  ^  c/,s, 

J  r  J   r  dn  J       dn 

where  the  integrals  are  line  integrals  along  the  curve  and  dr/dn  is  tlie  normal 
derivative  of  r,  is  the  angle  cp  subtended  by  the  curve  at  ;•  =  U.    Hence  infer  that 

rdhv^r  /-(Hogr  rdhiixr 

I  : —  (7.S  =  2  TT     or     I   ds  =  0     or     I   da  =  & 

Jq    dn  Jq    d)L  Jo    dn 

according  as  the  point  j-  =  0  is  within  the  curve  or  outside  the  curve  or  upon 
the  curve  at  a  point  where  the  tangents  in  the  two  directions  ai'e  inclined  at  the 
angle  0  (usually  it).  Note  that  the  fornuda  may  be  appiie(l  at  any  point  (f,  tj)  if 
r-  =  (I  —  x)-  +  (t;  —  //)-  where  (j,  y)  is  a  point  of  the  curve.  What  would  the  inte- 
gral give  if  aiiplied  to  a  space  curve? 

17.  Are  the  line  integrals  of  Ex.  10  of  the  same  type  i  P{f.  y)dx  +  Q{:c,  y)dy 

as  tho.se  in  the  text,  or  are  they  more  intimately  as.sociated  with  the  curve  '.'  Cf .  §  155. 

J-.0,  1  /I  0,1 

(,f  —  y)dH.  (13)  I        .eyds  along  a  riuht  line,  alontr  a  quad- 
1.0  ^-1,0 

rant,  along  the  axes. 

124.  Independency  of  the  path.  It  has  l)een  seen  that  in  ease  the 
integral  around  every  (dosed  patli  is  zer(^  or  in  case  the  integrand 
J'd.r  +  (I'/i/  is  a  total  ditt'erential,  the  integral  is  inde])endent  of  the 
path,  and  eonversely.    Hence  if 

]><lr  +   Q'h,,        tlKMl        -^  =  ]\  -;:"  =   (}, 

,,  ^■•'  ^!l 

c-F        cQ  c-F        cP  cP       cQ 

and  ^  ..    =  :  —  J  ^   „    =  ,-—  J         —  =  --  J 

CXCi/         C.r  Ci/C.r  C  ij  C  ij  CX 

provided  the  })avtial  derivatives  P'^  and  (l',.  are  continiioiis  functions.* 
It  remains  to  ])rove  the  converse,  namely,  that:  If  flw  tiro  jxirtial 
der'traflrcs  7*,^  (did   Q',.  (tri'  contlnKous  and  oiikiI,   flic   Intcgrdl 


L 


pdx  +  Qdij    with    y;  =  (^  (18) 


i.s  indcpcndinf  nf  the  jjafJi.  is  r:i'fo  tirmoid  c  i-loscd  jxtfli,  and  fhf,  quantlti/ 
Pdr  +  Q<h/  is  <t   fntdi  dijfn-rntinl. 

To  show  tliat  tlic  integral  of  l^d.r  -)-  Qjy  around  a  closed  ])ath  is  zero 
if  P,^  =  Q',.,  c(.)nsider  first  a  region  /'  su(di  that  any  point  (./•.  //)  of  it  may 

*  Sec  §  ."ii'.  Ill  particular  (ihscrve  the  comnu'iits  there  made  relative  to  differentials 
whit'li  are  or  which  arc  ijdt  exact.  Tliis  difference  corresponds  to  integrals  which  are 
anil  which  arc  not  iiiclciiciKlciit  of  the  i)ath. 


ox    SIMPLE    INTKCiRALS 


299 


be  reached  from  ('/,  h)  hy  following  the  lines  jj  =  l>  and  ,/•  =  .>■.    Then 
define  the  function  /■'(.'■,  //)  as 

F 

!'(,-,  y)=j  }'(■';  ^>)'^-'-  +r^^('''  y^'^y     (^^) 

for  all  })oints  of  that  rcyion  11.    Xow 

cF  cF  ,  c     C" 


^u 


But 


This  results  from  Leibniz's  rule  (4')  of  §119.  which  may  l)e  ai)])lied 
since  (}',.  is  l)y  hy})othesis  continuous,  and  from  th(^  assumption  Q',.  =  P'y 
Then  ^y,- 

Hence  it  follows  that,  within  the  region  specified,  Pdx  +  Qdij  is  tjie 
total  differential  of  the  function  F{.i-,  //)  defined  l)y  (10).  Hence  along 
any  closed  circuit  Avithin  that  region  A'  tlie  integral  of  Pdx  +  (id ij  is 
the  integral  of  dF  and  vanishes. 

It  remains  to  roinove  the  restriction  on  tlie  type  nf  retfion  within  wliich  the 
integral  around  a  closed  path  vanishes.  Consider  any  closed  path  f  which  lies 
within  the  region  over  which  P,^  and  C^,'  are  e(jual  continuous  functions  of  (j,  y). 
As  the  path  lies  wholly  within  II  it  is  jjossihle  to  rule  l\  so  tinely  that  any  little 
rectangle  which  contains  a  portion  of  the  jiath  shall  lie  wholly  within  11.  The 
reader  may  construct  his  own  figure,  jjossihly  with  reference  to  that  of  §  128,  where 
a  finer  ruling  would  be  needed.  The  path  ('  may  thus  be  surrounded  by  a  /.ig/.ag 
line  which  lies  within  7i.  Each  of  the  small  rectangles  within  the  zigzag  line  is  a 
region  of  the  type  aljove  considei-ed  and,  by  the  pi'oof  above  given,  the  iiUegral 
around  any  closed  curve  within  the  small  rectangle  nmst  be  zero.  Now  the  circuit 
C  may  be  reiilaeed  by  the  totality  of  small  circuits  consisting  either  of  the  i)erim- 
eters  of  small  rectangles  lying  wholly  within  C  or  of  portions  of  the  ciu've  C  and 
portions  of  the  perimeters  of  such  rectangles  as  contain  parts  of  C.  And  if  C  be  so 
replaced,  the  integral  around  C  is  resolved  into  the  sum  of  a  large  luunber  of  inte- 
grals about  these  small  circuits;  for  the  integrals  along  such  parts  of  the  small 
circuits  as  are  jiortions  of  the  perimeters  of  the  rectangles  oecui'  in  pairs  with  oppo- 
site signs.*  Hence  the  integral  around  C  is  zero,  where  C  is  any  circuit  within  A'. 
Henct'  the  iiUegral  of  Vdx,  +  (Idy  from  {(i.  h)  to  (x,  y)  is  independent  of  the  path 
and  delines  a  function  F{.r.  y)  of  which  Pihc  +  (lily  is  the  total  differential.  As 
this  function  is  continuous,  its  value  for  points  on  the  l)omulary  of  7.'  may  be  detined 
as  the  limit  of  F{,r.  y)  as  (,r,  y)  approaches  a  point  of  the  boundary,  and  it  may  thereby 
be  .seen  that  the  line  integral  of  (18)  around  the  lioundary  is  also  0  without  any  fur- 
ther restriction  than  that  P'^  and  Q'.  be  ecjual  and  continuous  within  the  boundary. 

*  Sec  Ex.  10  above.  It  is  well,  in  coniiectidu  witli  §§  li'.'^-bJ.").  to  read  carefully  the 
work  of  §§  44-45  dealing  with  varieties  of  regions,  redueibility  of  circuits,  etc. 


800  INTEGRAL  CALCULUS 

It  should  be  noticed  that  tJia  line  integnd 

f  ''pdx + (i<j<j  =  r  p(.>-,  h)dx  +  r  iux,  y),hj, 

J  (I,  h  Ja  Jli 


(19) 


ii-Jicn  P(Jx  +  Qdij  is  an  exuct  diffcrentlid,  that  is,  when  P,',  =  Q\.,  way  he 
('(•ahiafed  by  the  rule  glrcn  for  infi'(/ratln(j  an  exfU't  dlffefe7it'uij  (p.  209j, 
])rovided  tlic  patli  along  y  =  h  and  x  =  x  does  not  go  outside  the  region. 
If  tliat  path  should  cut  out  of  It,  sonie  other  method  of  evaluation  would 
be  recpiired.  It  should,  however,  be  borne  in  mind  that  Pdx  +  Qdy 
is  best  integrated  by  iiispe(.'tion  whenever  the  function  F,  of  wliich 
Pdx  +  Qdy  is  tlie  differential,  can  be  recognized  ;  if  F  is  multiple  valued, 
the  consideration  of  the  path  may  l)e  required  to  pick  out  the  par- 
ticular value  Avliich  is  needed.  It  nuiy  be  added  that  the  work  may  be 
extended  to  line  integrals  in  space  without  any  material  modifications. 
It  was  seen  (§  73)  that  the  conditions  that  the  complex  function 

Fi:'',  U)  =  ^  (■'■?  U)  +  ''^'(•'•,  U):  -  =  a-  +  iy, 

l)e  a  function  of  tlie  complex  variable  z  are 

a;  =  -i;  and  x:=y;.  (20) 

If  these  conditions  be  applied  to  the  expression  (13), 


J.I.  h 


I  Fix,  y)=    I        Xdx  -  Ydy  +  [  I         Ydx  +  Xdy, 

J  J„.  h  J.f.  h 

for  llic  line  iiili'gral  of  such  a  junction,  it  is  seen  that  they  are  pre- 
cisely tlie  conditions  (IS)  that  I'ach  of  tlic  line  integrals  entering  into 
the  coiu])lcx  line  integral  shall  be  independent  of  the  path.  Hence 
tJic  Infcgrdl  of  a  fii ncflon  of  a  coiiijdr.r  rarlohlt',  is  indt'pnident  of  the 
path  if  irdcgrafion  in  tin-  couqdcx  jdanc,  and  flic  intcyral  a ronnd  a 
(dosed  path  eanislies.  This  applies  of  course  only  to  simply  connected 
I'cgions  of  tlie  plane  throughout  which  the  derivatives  in  (20)  are  equal 
and  continuous. 

If  the  notations  of  vectors  in  three  dimensions  be  adopted, 

\   Xdx  +  Ydy  +  Zd::  =    /F.r/r, 


/' 


where  F  =  A'i  -f  ]'j  -f  Zk,         dx  =  \dx  +  yl y  -f  k'Av. 

In  the  particular  case  where  the  integi'and  is  an  exact  differential  and 
<:])e  integral  around  a  closed  })ath  is  zero, 

Xdx  -f-  Ydy  -f  Z,h:  =  Y .dx  =  dU  =  dX'\U, 


ox   SIMPLE   IXTEGEALS  301 

where  T  is  the  function  defined  by  the  integral  (for  V6'  see  p.  172). 
When  F  is  interpreted  as  a  force,  the  function  I'  =  —  IJ  such  that 

„  cV  cV  cV 

Y  =-\V     or     A'  =  -  — ,  }-  =  -  ^—  ,  z  =  --T- 

c.i-  cij  cz 

is  failed  the  potential  function  of  the  force  F.  Tlie  nrgafln'  nf  the 
tihj)e  of  the  jiotent'idl  function  is  the  farce  F  and  tlte  negatives  of  the 
partial  dcricatires  are  the  coniponent  forces  alomj  the  axes. 

If  the  forces  are  such  that  they  are  thus  derivable  from  a  potential  function, 
they  are  said  to  be  conservative.    In  fact  if 


and 


Thus  the  sum  of  the  kinetic  enersy  Imv-  and  the  potential  enerrn;^  T'  is  the  same 
at  all  times  or  positions.  This  is  the  principle  of  the  conservation  of  energy  for  the 
simple  case  of  the  motion  of  a  particle  when  the  force  is  conservative.  In  case  the 
force  is  not  conservative  the  integration  may  still  be  performed  as 


.4NF  =  -Tr, 

dt- 

in  '^'^.dr  =  -  (ir.V T'  =  -  dV, 

dt- 

1     m  —  'dx  — 
Jr,       dV^ 

m  dx  (?r  !  r,               !  ri 

~2  dt'  dt  f'~           ' 

or     -^  v^  +  T  1  =  _.  1,^-  +  T  0. 

|(rf-r|)=/^'F..r  =  T,', 


where  W  stands  for  the  work  done  by  the  force  F  (Uu-infj;  the  motion.  The  result  is 
that  the  cliange  in  kinetic  energy  is  eqv;al  to  the  worlv  done  l)y  the  force ;  but  dW 
is  then  not  an  exact  differential  and  the  work  nuist  not  be  regarded  as  a  function 
of  (.r.  y.  z).  —  it  depends  on  the  path.  The  generalization  to  any  number  of  particles 
as  in  §  123  is  inunediate. 

125.  The  conditions  that  /',^  and  ll',.  l)e  continuous  and  equal,  which 
insures  independence  of  the  ])at]i  for  the  line  integral  of  Pdx  +  Qdi/, 
need  to  be  examined  more  closely.    Consider  two  examples  : 

First  C Pdx  +  qdy  =  f    ~  ■'  -  dx  +     /    ~<ly , 

J  J  X-  +  y-  X-  +  y'- 

cP         y-  —  x-  cQ        y"  —  x- 


where 


cy      (.«•-  +  .'/'-)-  ex      {x-  +  y-y- 


It  appears  formally  that  P,^  =  Q'..    If  the  integral  be  calculated  around  a  square  of 
side  2  a  surrounding  the  origin,  the  result  is 


X  +  n  +  (idx        r  +  "    ady  ^  -r,  _  ^^/,.        ^  -  n  _  f,fiy       -^  r  '"■    ndx 

4 -.  +  I  o         ,  +  o  -    -  ,  +  o  -  -  .,  =  2  .,   ,      , 

-a  X-  +  «-      -'-<(  (I-  +  y      d+„  .r-  +  <i~      J+„   a-  +  y-         J-a  x-  +  a 

+  2  I        ^—  =  4  I —  =  4  -  =  2  TT  ?i  0. 

J- a  a'^  +  2/-        J- a  ^-  +  a-         2 


302  INTEGRAL  CALCULUS 

The  integral  fails  to  vanish  around  the  closed  path.  The  reason  is  not  far  to  seek, 
the  derivatives  P'  and  Q^  are  not  defined  for  (0,  0),  and  cannot  be  so  defined  as 
to  be  continuous  functions  of  {x,  y)  near  the  origin.    As  a  matter  of  fact 


I        — f- ^  =    I        d  tan  -1  •'  =  tan  -i 

J<,,  h    X-  +  y-      x-  +  y-      J  a,  h  X 


and  tan  -i  {y/x)  is  not  a  single  valued  function  ;  it  takes  on  the  increment  2  tt  when 
one  traces  a  patli  surrounding  the  origin  (§45). 
Another  illustration  may  be  found  in  the  integral 

/dz        r  dx  +  idy  _    r  xdx  +  ydy  ,    ■  f  —  V^^:  +  xdy 
z        J     X  +  iy         J      X-  +  y-  J        x^  +  y- 

taken  along  a  path  in  the  complex  plane.  At  the  origin  z  =  0  the  integrand  \/z 
becomes  infinite  and  so  do  the  partial  derivatives  of  its  real  and  imaginary  parts. 
If  the  integral  be  evaluated  around  a  path  passing  once  about  the  origin,  the 
result  is 

I    _  =  U  log  (x2  4-  ?/2)  +  i  tan  -i  ^  =2iri.  (21) 

Jo  z        \_2     '  irjfi,6 

In  tliis  case,  as  in  tlie  previous,  tlie  integral  would  necessarily  be  zero  about  any 
closed  path  which  did  not  include  the  origin  ;  for  then  the  con- 
ditions for  absolute  independence  of  the  path  would  be  satisfied. 
Moreover  the  integrals  around  two  different  ^laths  each  encircling 
the  origin  once  would  be  equal ;  for  the  paths  may  be  considered 
as  one  single  closed  circuit  by  joining  them  with  a  line  as  in  the 
device  (§  44)  iov  making  a  nuiltiply  connected  region  simply  con- 
nected, the  integral  around  the  complete  circuit  is  zero,  the  parts 
due  to  the  description  of  the  line  in  the  two  directions  cancel, 
and  the  integrals  around  the  two  given  circuits  taken  in  opposite  directions  are 
therefore  equal  and  opposite.  (Compare  this  work  with  the  nuiltiplc  valued  nature 
of  logz,  p.  161.) 

Suppose  in  general  that  P(.r.  //)  and  Q(x,  y)  are  single  valued  func- 
tions wliicli  liave  the  first  partial  derivatives  P,^  and  <.l',  continuous 
and  equal  over  a  region  7'  exeept  at  certain  points  .1,  B,  ■■■.  Surround 
tliese  points  with  small  circuits.  The  remaining  }iortion  of  Tt  is  such 
that  P'y  and  Q'  are  everyvrhere  equal  and  continuous;  but  the  r(\gion 
is  not  simply  connected,  that  is,  it  is  ])Ossil)lc  to  draw  in  tlic  region 
circuits  which  cannot  l)e  shrunk  down  to  a  })oint,  owing  to  the  fact 
that  the  circuit  may  surround  one  or  more  of  tlitj  regions  which  have 
heen  cut  out.  If  a  circuit  can  be  shrunk  down  to  a  point,  that  is,  if  it 
is  not  inexti'icably  wound  about  one  or  more  of  the  deleted  ])0]-tions, 
the  integi'al  around  the  circuit  will  vanish;  for  the  })revious  reasoning 
will  apply.  ])Ut  if  the  circuit  coils  about  one  or  more  of  the  deleted 
regions  so  that  the  attenqjt  to  shrink  it  down  leads  to  a  circuit  which 
consists  of  the  contours  of  these  I'l^gions  and  of  lines  joining  them,  the 
integral  need  not  vanish  ;  it  reduces  to  the  sum  of  a  number  of  integrals 


ox  SIMPLE  IXTEGEALS  303 

taken  aroiiiul  the  contours  of  the  deleted  portions.  If  one  circuit 
can  be  shrunk  into  another,  the  integrals  around  the  two  circuits  are 
equal  if  the  direction  of  description  is  the  same ;  for  a  line  connecting 
the  two  circuits  will  give  a  combined  circuit  which  can  bo  shrunk  down 
to  a  point. 

The  inference  from  these  various  observations  is  that  in  a  multiply 
connected  region  the  integral  around  a  circuit  need  not  be  zero  and 
the  integral  from  a  fixed  lower  limit  {(i,  h)  to  a  variable  upper  limit 
(./•,  ?/)  may  not  be  absolutely  independent  of  the  path,  but  may  be  dif- 
ferent along  tAvo  paths  which  are  so  situated  relatively  to  the  excluded 
regions  that  the  circuit  formed  of  the  two  paths  from  (rr,  h)  to  (x,  7) 
cannot  be  shrunk  down  to  a  })oint.    Hence 

^  (■'■>  1/)^   f  '  'l'<^'-'-  +  d'^'J,         K  =  Q'x  (generally), 

the  function  defined  l)y  the  integral,  is  not  necessarily  single  valued. 
Kevertheless,  any  two  values  of  Fi^x,  y)  for  the  same  end  point  will 
difter  only  by  a  sum  of  the  form 

F'i(?',  y)  -  Fi{^,  y)  =  wiA  +  ^"2-^2  +  •  • 

where  /j,  In,  •  •  •  ^i'^  the  values  of  the  integral  taken  around  the  con- 
tours of  the  excluded  r(>gions  and  Avherc;  ;//j,  w.^^,  .  .  .  are  positive  or 
negative  integers  which  represent  the  number  of  times  the  combined 
circuit  formed  from  the  two  paths  will  coil  around  the  deleted  regions 
in  one  direction  or  the  other. 

126.  Sir])pose  that  /(,-:)  =  X(x,  y)  +  lY{x,  y)  is  a  single  valued  func- 
tion of  z  over  a  region  R  surrounding  the  origin  (see  figure  above),  and 
that  over  this  region  the  derivative  /'(.v)  is  continuous,  that  is,  the 
relations  A',,'  =  —  F^  and  X',.  =  Y',  are  fulhlled  at  every  })oint  so  that 
no  points  of  11  need  ho  cut  out.    Consider  the  integral 

over  paths  lying  witliin  7*.  Tlie  function  f(rc)/z  Avill  have  a  contin- 
uous derivative  at  all  points  of  Jt  except  at  the  origin  ,'v  =  0,  Avhere  the 
denominator  vanishes.  If  then  a  small  circuit,  say  a  circle,  be  drawn 
about  the  origin,  the  function /(«)/«  will  satisfy  the  requisite  condi- 
tions over  the  region  which  remains,  and  the  integral  (22)  taken  around 
a  circuit  whi(,-h  does  not  contain  the  origin  will  vanish. 

The  integral  (22)  taken  around  a  circuit  which  coils  once  and  only 
once  about  the  origin  will  be  e(_[ual  to  the  integral  taken  around  the 


304  INTEGRAL  CALCULUS 

small  circle  about  the  origin.    ISTow  for  the  circle, 

where  the  assumed  continuity  of  f(n)  makes  \r](z)\  <  c  provided  the 
circle  about  the  origin  is  taken  sufficiently  small.    Hence  by  (21) 

p^  dz  =    r^  dz  =  2  7rlf{0)  +  $ 
Jo    "  Jq 

with  1^1  =  1    fld:^^     r  h.  L/^l  s  e   r\ld  =  2  tte. 

{Jq  "  Jq    "  Jn 


Hence  the  difference  between  (22)  and  2  7rlf(0)  can  be  made  as  small 
as  desired,  and  as  (22)  is  a  certain  constant,  the  result  is 


X 


^dz  =  2'7rlf(0).  (23) 


A  function  /(.")  which  has  a  continuous  derivative  /'(.")  at  every 
point  of  a  region  is  said  to  be  annli/tle  over  that  region.  Hence  if  the 
I'egion  includes  the  origin,  the  value  of  the  analytic  function  at  the 
origin  is  given  by  the  formida 

•^^^)  =  2^-   r^'^-  (23') 

Jq 

where  the  integral  is  extended  over  any  (drcuit  lying  in  the  region  and 
passing  just  once  al)Out  the  origin.  It  follows  likewise  that  if  z  =  a  is 
any  point  within  the  region,  then 

f{n)  =  ^.   f^^^dz,  (24) 

Jo 

where  the  circuit  extends  once  around  the  ])oint  a  and  lies  wholly  Avitliin 
the  region.    This  inn)ortant  result  is  due  to  ('aucliy. 

A  more  (-onvenient  form  of  (24)  is  obtained  by  letting  t  =  z  repre- 
sent the  value  of  z  along  the  circuit  of  integration  and  then  writing 
a  =^  z  and  regarding  z  as  variable.    Heiu^e  Cauchy's  Integral : 


This  states  that  if  (imj  rirrult  he  drmrn  in  flic  refjum  over  irhlch  f{z) 
is  anah/fir,  flic,  value  "f  f(z)  at  all  //oints  iritJiln  flint  cirn/U  iiKiij  he,  eijt- 
tained  hij  eealiKitlnr/  C<nie]iijs  Inte<jral  (25).   Thus  f(z)  may  be  regarded 


ON   SIMPLE   INTEGRALS 


305 


as   defined  by  an   intejj^ral   containing  a  parameter  z ;  for  many   pur- 
poses this  is  convenient.    It  may  be  remarked  that  Avhen  the  values  of 
fiz)  are  given  along  any  circuit,  the  integral 
may  be  regarded  as  defining /(,t;)  for  all  points 
within  that  circuit. 

To  iind  the  sKccc'ixlve  derlrati/'es  of  /(z),  it 
is  merely  necessary  to  differentiate  with  respect 
to  z  under  the  sign  of  integration.  The  condi- 
tions of  continuity  which  are  required  to  justify 
the  diiferentiation  are  satisfied  for  all  points  z 
actually  Avithin  the  circuit  and  not  upon  it.  Then 

(n 


/'(-) 


1    r  .rV) 

2  TT.-   /(/-,-: 


f' 


■,  r 


>(.) 


1)!  r  ,m 


•Jo 


dt. 


As  the  differentiations  may  be  performed,  these  formulas  sIioav  that  an 
(i7inlytlc  function  lies  contmuous  deriiuitlres  of  oil  orders.    The  definition 
of  the  function  only  required  a  continuous  first  derivative. 
Let  a  be  any  particular  value  of  ,',-  (see  figure).    Then 

1  1  11 


{^t  —  a)  —  {^z  —  a)        t  —  a  z  —  (C 


1  - 


1  + 


^^(. 


'^T 


(^  -  '0'- 


+ 


(z  —  a)" 
(t  —  a)" 


fit) 


(t-nf 


+  ^.  r^.-.)-^^'^^  +  --+^  {iz-ay-^j^^dt^n 
with  n^,  =  ^  f^^^^^       '        -^'^'^  - 


t  —  a 


t  —  a 


Now  t  is  the  variable  of  integration  and  z  —  «  is  a  constant  with  respect 
to  the  inte<rration.    Hence 


/(^)  ^/CO  +  (^  -  ^O/'l'O  +  ^^^:7T^/"('0 


+  •••  + 


(z  -  ay 


(26) 


This  is  Taylor's  Foi'miila  for  a  function  of  a  complex  variable. 


306  INTEGKAL  CALCULUS 

EXERCISES 

1.  If  P'y  =  Q^,,  Q'^  —  R',^^  li'^  =  P'^  and  if  these  derivatives  are  continuous,  show 
that  Pdx  +  Qdy  +  Rdz  is  a  total  differential. 

2.  Show  that       I        P  {X:,  V-,  ir)dx  +  Q(x,  y,  a)d)/,  where  C  is  a  given  curve, 

defines  a  continuous  function  of  <(-,  tlie  derivative  of  which  may  be  fomid  by  differ- 
entiating under  the  sign.  What  assumptions  as  to  the  continuity  of  P,  Q,  P^,  Q^ 
do  you  make  ? 

o.  If   logz=    I     — =    I \-  I  I         be    taken   as  the 

Ji    z       J  1,0       x^  +  y'-  J  1,0         X- +  y'^ 

definition  of  log  z,  draw  paths  which  make  log  (J  +  },  V—  3)  =  jTri,  2  J  tt/,  —  1  jj  tt/. 

/.s  3^  —  1 

4.  Study    I        ~  with  especial  reference  to  closed  paths  which  surround  +  1 

Jo    z^  —  1 
—  1,  or  both.   Draw  a  closed  path  surrounding  both  and  making  the  integral  vanish. 

5.  If /(z)  is  analytic  for  all  values  of  z  and  if  \f(z)  \  <  7r,  show  that 

/(z)  -  /(o)  =  r  ./-(o  [^  ^  -  -  J]  ./<  =  r  -f^^^^^  du 

Jo        \_i  —  z      i\  Jo{l  —  z)t 

taken  over  a  circle  of  large  radius,  can  be  made  as  small  as  desired.  Hence  infer 
that/(2)  nuist  be  the  constant/(z)  =/(0). 

6.  If  G  (z)  =  (7„  4-  a-jZ  +  •  •  •  +  <i„'^"  i>^  i>  polynomial,  siiow  that/(z)  =  ^/G  (r)  nuist 
be  analytic  over  any  region  which  docs  not  include  a  root  of  G  (z)  =  0  either  witliin 
or  on  its  boundary.  Show  that  the  assumption  tliat  (r{z)  =  0  has  no  roots  at  all 
leads  to  the  conclusion  that /(z)  is  constant  and  equal  to  zero.  Hence  infer  that 
an  algebraic  equation  has  a  root. 

7.  Show  that  the  absolute  value  of  the  remainder  in  Taylor's  Fonnula  is 

,^,,         \z-a\"     r  f(t)dt i        1     v   ML^ 

2-w        Jo{t  —  a)"  {t  —  z)\       •2Tr  p"  p  —  r 

for  all  points  z  within  a  circle  of  radius  r  about  a  as  center,  wlieii  p  is  (lie  radius 
of  the  largest  circle  concentric  with  a  which  can  be  drawn  within  the  circuit  alxnit 
which  tlie  integral  is  taken,  M  is  the  maxinuun  value  of  f{t)  upon  the  I'ircuit,  and 
L  is  the  length  of  tlie  circuit  (figure  above). 

8.  Examine  for  independence  of  path  and  in  case  of  independence  integrate: 

(a)    I  x-i/dx  +  xy-ili/,         {[i)    j  xy-dx  +  x-ydi/,         (7)    j  xih/ +  ydx, 
(5)     /   {x~  +  xy)dx +  {//-  + xy)dy,         (e)     |   // cos  jv/// +  '  //- sin  ^•(Zx. 

9.  Find  the  conservative  forces  and  the  iiotential : 


(/3)   X  =  -  nx.  ¥=.-  »//.  (7)   X  =  V.r.  Y  =  y/x. 


ox  SIMPLE  INTEGRALS  307 

10.  If  li  {)•,  <p)  and  #(/•,  (p)  are  the  component  forces  resolved  along  the  radius 
vector  and  perpendicular  to  the  radius,  show  that  clW  =  Udr  +  i-^d<f)  is  the  differ- 
ential of  work,  and  express  the  condition  that  the  forces  li,  <I>  be  conservati'/e. 

11.  Show  that  if  a  particle  is  acted  on  by  a  force  E  =  —/('')  directed  toward 
the  origin  and  a  function  of  the  distance  from  the  (jrigin,  the  force  is  conservative. 

12.  If  a  force  follows  the  Law  of  Nature,  that  is,  acts  toward  a  point  and  varies 
inversely  as  the  square  r-  of  the  distance  from  the  point,  show  that  the  potential 
is  —  k/r. 


13.  Troni  the  results  F  =  —  V  T  or  T'  =  -    C F<h  =    C A'dx  +  Ydij  + 


Zdz  show- 


that  if  T\  is  the  potential  of  Fj  and  T'.^  of  F.,  then  V  =  Y^  +  1'.  will  be  the 
potential  of  F  =  F,  +  F.,.  that  is,  show  that  for  conservative  forces  the  addition  of 
l^titcntials  is  e(iuivalent  to  the  parallelogram  law  for  adding  forces. 

14.  If  a  particle  is  acted  on  by  a  retarding  force  —  ky  proportional  to  the 
velocity,  show  that  11  =  \  kv-  is  a  function  such  that 

-  ~  =  —  ki\,:         —  =  —  kv,f,         —  =  —  A-y^, 

(l-'.c  fl'.v  Cl'j 

d]V=  -  kv.dT  =  -  k{i\,d.c  +  v,,di/  +  vMz). 
Here  E  is  called  the  dissipative  function  ;  show  the  force  is  not  conservative. 

15.  rick  out  the  integrals  independent  of  the  path  and  integrate: 

(a)     r  t/zdx  +  .rzdij  +  xydz,       {(S)    j  ijd.r/z  +  ■'■dtj/z  —  xijdz/z'^, 
(7)   J  -'-l/z  (d.r  +  dii  +  (?-).  ( o)   J  log  (.r //)  dx  +  xdy  +  ijdz. 

16.  Obtain  logarithmic  forms  for  the  inverse  trigonometric  functions,  analogous 
to  those  for  the  inverse  hyperbolic  functions,  either  algebraically  or  by  considering 
the  inver.se  trigonometric  functions  as  defined  by  integrals  as 

dz  .      ,  r~-       dz 


1  r~     dz  .      .  p-       dz 

t/U       1   -\- Z-  Jo       -y^l  r- 


17.  Integrate  these  functions  of  the  comi)lex  variable  directly  according  to  the 
rules  of  integration  for  reals  and  determine  the  values  of  the  integrals  by 
substitution : 

(a)   £^'zv-ih,  iiS)   £"cosiizdz,  (7)   f^^'''0+z-^rhlz, 

^'^  -Vl-2-  ^'     Z\Z--l  '^-l  Vl-l-2- 

In  the  case  of  nudtipic  valued  functions  mark  two  different  paths  and  give  two  values. 

18.  Can  the  algorism  of  integration  by  parts  be  apjjlied  to  the  definite  (or  indeti- 
nite)  integral  of  a  function  of  a  complex  variable,  it  being  understood  that  the 
integral  must  Vie  a  line  integral  in  the  complex  plane?  Consider  the  proof  of 
Taylor's  rormula  by  integration  by  parts,  p.  57,  to  ascertain  whether  the  proof  is 
valid  for  the  complex  plane  and  what  the  remainder  means. 


308  INTEGRAL  CALCULUS 

19.  Suppose  that  in  a  plane  at  ?•  =  0  there  is  a  particle  of  mass  in  ■which  attracts 
accordiiiic  to  the  law  F  =  m/r.  Show  that  the  potential  is  V  =  mloi^r,  so  that 
F  =  —  VT".  The  induction  or  fliu  of  the  force  F  outward  across  the  element  ds  of 
a  curve  in  the  plane  is  by  definition  —  Fcos(F,  n)ds.  By  reference  to  Ex.  16, 
p.  297,  show  that  the  total  induction  or  liux  of  F  across  the  curve  is  the  line  integral 
(along  the  curve) 

—    I   Fcos(F,  )()(/.s  =  ;/i  j    ^^^  (i,s  =    /    — ds  ; 

J  J       dn  J    dn 

I  r  ^      ,^    -  ,        i^  r  dV 

Itt  ^o  dn 


and  m  =  - —   |  F  cos  (F,  n)  ds  =  —   I    -^—  ds. 


where  the  circuit  extends  around  the  point  r  =  0,  is  a  fornnda  for  obtaining  the 
mass  m  within  the  circuit  from  the  field  of  force  F  which  is  set  up  by  the  mass. 

20.  Suppose  a  luunber  of  masses  w, .  »/.,.  •  ■  • .  attracting  as  in  Ex.  l!t,  are  situated 
at  points  (^j,  Tjj),  (^.,,  V-^)-  •  ■  ■  in  the  i>hirie.    Let 

F  =  F^  +  F,  +  •  •  • ,  V  =  V^  +  v.,  +  ■  ■  ■ ,  Vi  =  m,-  log  [i^,-  -  xf  +  (t?,-  -  yf]l 

be  the  force  and  potential  at  (.r,  //)  due  to  the  masses.   Show  that 

:^'  f  FeoHF,n)ds  =  -Ly  f  '^ds=y'n,  =  M, 

ZTT  Jo  ZTT  ^~^  •JQ  dn  -^^ 

where  2  extends  over  all  the  masses  and  2'  over  all  the  masses  within  the  circuit 
(none  lieing  on  the  circuit),  gives  the  total  mass  3f  witliin  tlie  circuit. 

127.  Some  critical  comments.  In  the  discussion  of  line  int(\t,Tals 
and  in  the  future  discission  of  double  integrals  it  is  necessary  to  speak 
fre(jiieutly  of  curves.  For  the  usual  problem  the  intuitive  conception 
of  a  curve  suthces.  A  curve  as  ordinarily  conceived  is  continuous,  has 
a  continuously  turning  tangent  line  except  perluqis  at  a  finite  ntuubcr 
of  angular  points,  ami  is  cut  by  a  line  parallel  to  any  giveii  direction  in 
oidy  a  finite  ]uunl)er  of  points,  exce])t  as  a  portion  of  the  curvt^  may 
(U)iiicide  with  such  a  line.  The  ideas  of  lengtli  and  area  are  also  aj)])li- 
cable.  For  those,  however,  who  are  interested  in  more  than  the  intuitive 
])resentation  of  the  idea  of  a  curve  ami  some  oi'  the  matters  therewith 
coniu'cted,  the  following  sections  are  offered. 

If  (p  (t)  and  ^  (t)  are  two  single  valued  real  functions  of  the  real  variable  t  defined 
for  all  values  in  the  interval  t^y  =  t  ^  t^.  the  pair  of  t'(]uations 

x  =  4>{t).         !/  =  ^{t).         /„^/^/,.  (27) 

will  be  said  to  define  a  rune.  If  0  and  \p  are  continuous  functions  of  t,  the  curve 
will  be  called  continuous.  If  (p{t^)  =  0(/,,)  and  •/'(/,)  =  ^p  (^,,).  so  that  the  initial  and 
end  points  of  the  curve  cfdncide.  the  curve  will  lie  called  a  closed  curve  i)rovided 
it  is  continuous.  If  there  is  no  other  iiair  of  values  t  and  V  whicii  make  both 
(j)(t)  —  (p{t')  and  '^(0  ~  ^(f).  the  curNc  will  be  calJi'd  shujilc:  in  ordinary  language, 
the  curve  does  not  cut  itself.  If  /  describes  the  iiiter\al  from  /,,  to  [^  cuntiuueusly 
and  cnnstaiUly  in  tlie  same  sense,  the  jiiiint  i.e.  //)  will  be  said  td  desci-ilie  the  curve 
in  a  given  sense  ;  the  (ipjMisite  sense  can  be  had  by  allowing  t  to  describe  tlie  interval 
in  the  opposite  direction. 


ox   SIMPLE   INTEGRALS  309 

Let   the   interval    t^y  ^  t  ^  t^   be  divided   into  any  number  n  of  subintervals 
A^t,  A.,t,  ■  ■  •,  A„t.    There  will  be  n  corresponding  increments  for  x  and  y, 

A,/.  A.,J",  •  •  ■ ,  A„j,     and     A^y.  A.,y,  •  •  • ,  A„y. 


Then    A,-'-  =  a  (A,j-)-  +  {A,y)-  =1A,-J-|  +  |A,-?/|,         ] A/x|  ^  A,c,         |A,-2/|  ^  A,-c 
are  obvious  ine(]ualities.     It  will  be  necessary  to  consider  tlie  tliree  sums 


a,  =^   A„r:.  <r,  =^|A,-2/],  <r,  =2^A,C  =:^  V(A,.,-)-  +  {A^yf. 

1  1  11 

Fur  any  division  of  tlie  interval  from  t^  to  t^  each  of  these  sums  has  a  definite 
positive  value.  When  all  possible  modes  of  division  are  considered  for  any  and 
e\ery  value  of  ;/.  the  sums  o-j  will  form  an  infinite  set  of  numbers  which  may  be 
either  limited  or  unlimited  above  (§22).  In  case  the  set  is  limited,  the  ujiper 
frontier  of  the  set  is  called  the  variation  of  x  over  the  curve  and  the  curve  is  saiil 
to  be  of  limited  variation  in  x;  in  case  the  set  is  unlimited,  the  curve  is  of  unlimited 
variation  in  x.  Similar  observations  for  the  sums  o-.,.  It  may  be  remarked  that  the 
geometric  conception  corresponding  to  the  variation  in  x  is  the  sum  of  the  projec- 
tions of  the  curve  on  the  j-axis  wlien  the  .sum  is  evaluated  arithmetically  and  not 
algebraically.  Thus  tlie  variation  in  y  for  the  curve  y  =  sinx  from  0  to  2  7r  is  4. 
The  curve  y  =  sin(l//)  between  these  .same  limits  is  of  unlimited  variation  in  y. 
In  both  cases  the  variation  in  x  is  2  tt. 

If  both  the  .sums  (r,  and  tr.,  have  upper  frontiers  L^  and  Z.,,  the  .sum  o-g  will  have 
an  upper  frontier  i.,  ^  -L,  +  L., :  and  conver.sely  if  ff.,  has  an  upper  frontier,  both 
(7j  and  (T.,  will  have  upper  frontiers.  If  a  new  point  of  division  is  intercalated  in  A,-^, 
the  sum  o-j  cannot  decrease  and,  moreover,  it  cannot  increa.se  by  more  tlian  twice 
the  oscillation  of  x  in  the  interval  A,^    For  if  Ai,x  +  Ao,x  =  A,J,  then 

fAi,-.f  I  +  \A.;X,  s  ;A,-.f  i.  |Ai,-x;  +  |Ao,-.r;  ^  2  (.V,-  -  ?«,). 

Here  Ant  and  Ao,i  are  the  two  intervals  into  which  A,^  is  divided,  and  3/",-  —  ??;,-  is  the 
oscillation  in  the  interval  A,^  A  similar  theorem  is  true  for  a.,.  It  now  remains  to 
show  that  if  the  interval  from  t^^  to  t^  is  divided  sutficieiitly  fine,  the  .sums  o-j  and  a.. 
will  diiSer  bj"  as  little  as  desired  from  their  frontiers  i^  and  Z.,.  The  proof  is  like 
that  of  the  .similar  problem  of  §  28.  First,  the  fact  that  L^  is  the  frontier  of  o-,  shows 
that  some  method  of  division  can  be  found  so  that  L^  —  a-^  <  le.  Suppose  the  num- 
ber of  points  of  divi.sion  is  n.  Let  it  next  be  a.ssumed  that  (p{t)  is  continuous;  it 
niiLSt  then  be  uniformly  contimious  (§25),  and  hence  it  is  possible  to  find  a  5  .so 
small  that  when  Ait  <  5  the  oscillation  of  x  is  ,V,-  —  ?»,-  <  e/^in.  Consider  then  any 
nieth(jd  of  division  for  which  A,-^  <  5,  and  its  sum  ff[.  The  superposition  of  the  former 
division  with  ?i  points  upon  this  gives  a  sum  a['  =  a[.  But  cr'^  —  a[<2  ne/A  n  =  J  c, 
and  a'^  ^  a-^.  Hence  L^  —  a'^  <  I  e  and  L^  —  a[<e.  A  similar  demonstration  may 
be  given  for  cr„  and  Z„. 

To  treat  the  sum  o-g  and  its  upper  frontier  L^  note  that  here,  too,  the  intercalation 
of  an  additional  point  of  division  cannot  decrease  o-g  and,  as 


A/(Ar)-  +  (A.'/)-^:A.r|  +  |A//|. 

it  cannot  increase  o-g  by  more  than  twice  the  sum  of  the  oscillations  of  x  and  y  in 
the  interval  At.  Hence  if  the  curve  is  continuous,  that  is,  if  both  x  and  y  are  con- 
tinuous, the  division  of  the  interval  from  i^  to  <j  can  be  t.iken  .so  line  that  a^  .shall 


310  INTEGRAL  CALCULUS 

differ  from  its  upper  frontier  ig  by  less  than  any  assigned  quantity,  no  matter  how 
small.  In  this  case  i^  =  s  is  called  the  length  of  the  curve.  It  is  therefore  seen  that 
the  necessai-y  and  sufficient  condition  tltat  any  continuous  curve  shall  have  a  length  is 
that  its  Cartesian  coordinates  z  and  y  shall  both  be  of  limited  variation.  It  is  clear  that 
if  the  frontiers  L^{t),  L„{t).  L.^{t)  from  ^^  to  any  value  of  t  be  regarded  as  functions 
of  i,  they  are  continuous  and  nondecreasing  functions  of  i,  and  that  L.^{t)  is  an 
increasing  function  of  t ;  it  would  therefore  be  possible  to  take  s  in  place  of  t  as 
the  parameter  for  any  continuous  curve  having  a  length.  ^Moreover  if  the  deriva- 
tives x'  and  y'  of  x  and  y  with  respect  to  t  exist  and  are  continuous,  the  derivative  s' 
exists,  is  continuous,  and  is  given  Ity  the  usual  fornuda  .s'  =  Vj-'-  +  y''^.  Tliis  will 
be  left  as  an  exercise ;  so  will  the  extension  of  these  considerations  to  three 
dimensions  or  more. 

In  the  sum  x-^  —  x^  =  2A,-x  of  the  actual,  not  absolute,  values  ()f  A,-.r  there  may 
be  both  iJO'^^itive  and  negative  terms.  Let  tt  be  tlie  sum  of  the  positive  terms  and 
V  be  the  sum  of  the  negative  terms.    Then 

Xj  —  Xy  =  TT  —  ;>,  (Tj   =  TT  +    V,  2  TT  =  Xj  —  X|j  +   ""i,  2  V  =:  X ^  —  X^   +  (T^. 

As  ff^  has  an  upper  frontier  L^  when  x  is  of  limited  variation,  and  as  x^  and  x^  are  con- 
stants, the  sums  ir  and  v  have  ixpper  frontiers.  Let  these  be  II  and  X.  dmsidered 
as  functions  of  i,  neither  n(i)  nor  N(^)  can  decrease.  Write  x{t)  =  x,^  -\-  Hit)—  'S{t). 
Then  the  function  x{t)  of  limited  variation  has  been  resolved  into  the  difference  of 
two  functions  each  of  limited  variation  and  nondecreasing.  As  a  linuted  non- 
decreasing  function  is  integrable  (Ex.  7,  p.  5-4),  this  shows  that  a  function  is  inttgralile 
over  any  irderval  over  which  it  is  of  limited  variation.  That  the  difference  x  =  x"  —  x' 
of  two  limited  and  nondecreasing  functions  must  be  a  functi(jn  of  limited  variation 
follows  ivom  the  fact  that '  Ax  ^  !-^"l  +  ;-^-'''|-    Lurtherniore  if 

X  =  Xy  +  n  -  X     be  written     x  =  [x^  +  II  +  \xj+  t  -  t,,]  -  [X  +  |Xy!  +  <  -  y , 

it  is  seen  that  a  function  of  limited  variation  can  be  regarded  as  tJie  difference  of  tv:o 
positive  functions  whicli  are  constantly  iucrca,-sing,  and  tliot  Uieve  functions  are  con- 
tinuous if  the  given  function  x  (t)  In  continuous. 

Let  the  curve  C  detined  by  tlie  equations  x  =  0(0-  i/  =  '/' (0-  ^o  —  '  —  ^-  ^^ 
continuous.    Let  r{x.  y)  be  a  continuous  function  of  (x.  //).    Form  the  sum 

^  r (t^■ .  -n;) A,x  =^r (^i ,  t;,-) a,-x" -^r i ^s- .  7,,) a,x'.  (28) 

wliere  A^x,  A.^x,  ■  •  ■  are  the  increments  corresponding  to  Ajf.  X,t.  ■  ■  ■ .  where  (^,-.  tj;) 
is  tlie  point  on  the  curve  which  corresponds  to  some  value  of  t  in  A,/,  wliere  x  is 
assumed  to  be  of  limited  variation,  and  wliere  x"  and  x'  are  tw(_i  continui;>iis  increas- 
ing functions  whose  difference  is  x.  As  x"  (or  x')  is  a  continuous  and  constantly 
increasing  function  of  t.  it  is  true  inversely  (Ex.  10.  p.  4.y)  that  t  is  a  continuous  and 
constantly  increasing  function  of  x"  (or  x').  As  P{x.  y)  is  continuous  in  (x.  //).  it 
is  continuous  in  t  and  also  in  x"  and  x'.  Xow  let  A,-^  =;  0  :  tlu-n  A,x"  =  0  and 
A,r'  =  0.    Also 

liiii V  PA/.'-"  =  f''Pd.r     and     lim V  P, A,-x' =   C'^Tdx'. 

•'O  ''u 

'I'iie  limits  exist  ami  arc  intt-iirals  simply  because  P  is  contiiuu.ius  in  x"  or  in  x'. 
Hence  the  nuul  on  the  It  ft  <f  (2!-i)  h<is  a  limit  ditd 

limV  I'Xx  -^      f   '  Pdx  =   r''  Pdx"  -  f^'J^'dx' 


ox   8I.MPLE   INTEGRALS 


311 


may  he  defined  as  the  line  integral  of  P  along  the  curve  C  of  limited  variation  in  z. 
The  a.s.suiiiption  that  y  is  of  limited  variation  and  that  Q{x,  y)  is  continuous  would 
lead  to  a  corresponding  line  integral.  The  assumption  that  both  x  and  y  are  of  limited 
variation,  that  is,  that  the  curve  is  rcctifiable,  and  that  P  and  Q  are  continuous  icould 
lead  to  the  existence  of  the  line  hdegral 


J"^'"''P{x,  y)dx+  Q{x,  y)dy. 


A  considerable  theory  of  line  integrals  over  general  rectifiable  curves  may  be  con- 
structed.   The  subject  will  not  be  carried  further  at  this  point. 

128.  The  (luestiou  of  the  area  of  a  curve  retiuires  careful  consideration.  In  the 
first  place  note  that  the  intuitive  closed  i)lane  curve  which  does  cut  itself  is  intui- 
tively believed  to  divide  tlie  plane  into  two  regions,  one  interior,  one  exterior  to  the 
curve  ;  and  these  regions  have  the  property  that  any  two  points  of  the  .same  regi(jn 
may  be  connected  by  a  continuous  curve  which  does  not  cut  the  given  curve, 
whereas  any  continuous  curve  which  coiniects  any  point  of  one  region  to  a  point 
of  the  other  nuist  cut  the  given  curve.  The  first  <iuestion  which  arises  with  regard 
to  the  general  closed  simple  curve  of  page  308  is  :  Does  .such  a  curve  divide  the  plane 
into  just  two  regions  with  the  properties  indicated,  that  is,  is  there  an  interior  and 
exterior  to  the  curve  ?  The  ansicer  is  affirmative,  but  the  i>ronf  is  somewhat  ditticnlt  — 
not  becau.se  the  .statement  of  the  problem  is  involved  or  the  proof  replete  with 
advanced  mathematics,  but  rather  because  the  statement  is  .so  simple  and  elemen- 
tary that  there  is  little  to  work  with  and  the  proof  therefore  reijuires  the  keenest 
and  most  tedious  logical  analy.sis.  The  theorem  that  a  closed  .simi)le  plane  curve 
has  an  interior  and  an  exterior  will  therefore  be  a.ssumed. 

As  the  functions  x{t),  y  (t)  which  define  the  curve  are  continuous,  they  are  lim- 
ited, and  it  is  possilile  to  draw  a  rectangle  with  sides  x  =  (t.  x  =  h.  y  =  '".  //  =;  d  so 
as  entirely  to  surround  tlie  curve.  This  rectangle  may  next  be  i-ulcd  with  a  num- 
lier  of  lines  i)arallel  to  its  sides,  and  thus  be 
divided  i]ito  .^mailer  rectangles.  These  little  rec- 
tangles may  be  divided  intr>  three  categories,  those 
outside  the  curve,  those  inside  the  curve,  and 
thosi'  up(.)n  the  curve.  B}'  one  upon  the  curve  is 
meant  one  which  has  .so  nmch  as  a  .'tingle  point 
fif  its  perimeter  or  interior  upon  the  curve.  Let 
.1.  .1,-,  ^l,,,  .le  denote  the  area  of  the  large  rec- 
tangle, the  .'<um  of  the  areas  of  the  small  rectan- 
gles, which  are  intei-ior  to  the  curve,  tin-  sum  of 
the  areas  of  those  upon  tlie  curve,  and  tlie  sum  of 
those  exterior  to  it.  Of  course  .1  =.l,-f  J„  +  .l,.. 
Now  if  all  methods  of  ndina-  be  cinsidcreil.  the 

qtu^ntities  -1,-  will  have  an  upper  frontier  L,-.  the  quantities  ylg  will  have  an  upper 
frontier  L^.  and  the  quantities  A„  will  have  a  lower  frontier  ?„.  If  to  any  method 
of  ruling  new  rulings  lie  a<liled.  the  (luaiit  ilies  A,-  and  .!,>  become  .1^  and  ^l'  with 
the  conditions  ,1,'  s  ^-1^.,  A'^,  ^  .1,,.  and  hence  J,',  ^  vi„.  From  this  it  follows  that 
A  =  L;  +  i„  +  L, .  For  let  there  be  three  modes  of  ruling  which  for  the  I'espective 
cases  .1;.  A,,.  A„  make  these  three  (luaiiTiries  difft^r  from  their  froiUiers  T;.  L,,.  l„ 
by  less  than  I  e.  Then  the  superposition  of  the  tliree  systems  of  ridings  gi\-es  rise 
to  a  rulinu:  for  which  A'-.  A',.  A'  nnist  differ  from  the  frontier  values  bv  less  than 


312  INTEGRAL  CALCULUS 

If,  and  hence  the  sum  Zj+/u+  L,.,  which  is  constant,  dilYers  from  the  constant  A 
by  less  tiian  e,  and  nuist  tlierefore  be  ecjual  to  it. 

It  is  now  possible  to  define  as  the  {qualified)  arvaa  nf  llie  curve 

Li  =  inner  area.         /«  =  area  on  the  curve.         i,  +  /„  =  total  area. 

In  the  case  of  curves  of  the  sort  intuitively  familiar,  the  limit  /„  is  zero  and 
Li  =  A  —  Lg  becomes  merely  the  (unqualified)  area  bounded  by  the  curve.  The 
question  arises  :  Does  the  same  hold  for  the  general  curve  here  tinder  discu.ssion  ? 
This  time  the  ansiner  is  negative ;  for  there  are  curves  which,  though  closed  and 
simple,  are  still  so  sinuous  and  meandering  that  a  finite  area  /„  lies  upon  the  curve, 
that  is,  there  is  a  finite  area  so  bestudded  with  points  of  tlie  curve  that  no  part  of 
it  is  free  from  points  of  the  curve.  This  fact  again  will  be  left  as  a  statement  with- 
out proof.   Two  further  facts  may  be  mentioned. 

In  the  first  place  there  is  applicable  a  theorem  like  Theorem  21,  p.  51,  namely: 
It  is  possible  to  find  a  number  5  so  small  that,  when  the  intervals  between  the 
ridings  (both  sets)  are  less  than  5,  the  sums  A„.  Ai.  A^  differ  from  their  frontiers 
by  less  than  2e.  For  there  is,  as  seen  above,  some  methud  of  ruling  such'  that  these 
sums  differ  from  their  frontiers  by  less  than  e.  Moreover,  the  adding  of  a  single 
new  ruling  cannot  change  the  .sums  by  more  than  AD.  where  A  is  the  largest  inter- 
val and  D  the  largest  dimension  of  the  rectangle.  Ileiu-e  if  the  tutal  munlicr  ui 
intervals  (both  sets)  for  the  given  method  is  -Vand  if  5  lie  taken  less  fhan  e/XAI). 
the  ruling  obtained  by  superpo.sing  the  given  ruling  upon  a  ruling  where  the  inter- 
vals are  le.ss  than  S  will  be  such  that  the  sums  differ  from  the  given  ones  by  less 
than  e,  and  hence  the  ruling  with  intervals  less  than  5  can  only  give  rise  to  sums 
which  differ  from  their  frontiers  by  less  than  2e. 

In  the  second  place  it  should  be  ob.served  that  the  limits  i,-,  l,,  have  been  obtained 
by  means  of  all  possible  modes  of  ruling  where  the  rules  were  parallel  to  the  x-  and 
2/-axes,  and  that  there  is  no  a  priori  a.ssurance  tliat  these  .same  linuts  would  have 
been  obtained  by  rulings  parallel  to  two  other  lines  of  the  plane  or  b}'  covering  the 
]ilane  with  a  network  of  triangles  or  hexagons  or  other  figures.  In  any  thorough 
treatment  of  the  subject  of  area  such  matters  would  have  to  be  discussed.  That 
the  discussion  is  not  given  here  is  due  entirely  to  the  fact  that  these  critical  com- 
ments are  given  not  so  much  with  the  desire  to  establish  certain  theorems  as  with 
I  he  aim  of  showing  the  reader  the  sort  of  <iiiestions  which  come  up  for  considera- 
I'nn  in  the  rigorous  treatment  of  such  elementary  matters  as  "the  area  of  a  jilane 
curve,''  which  he  may  have  thought  he  "knew  all  about. "' 

It  is  a  connnon  intuitive  conviction  that  if  a  region  like  that  formed  In'  a  square 
be  divided  into  two  regions  by  a  contimious  curve  which  runs  across  the  square 
from  one  point  f)f  the  bcjundary  to  another,  the  area  of  the  S(|uare  and  the  sum  of 
the  areas  of  the  two  parts  into  which  it  is  divided  are  ecjual.  that  is.  the  curve 
(counted  twice)  and  the  two  portions  of  the  iierimeter  of  the  square  form  two 
siuqile  closed  curves,  and  it  is  expected  that  the  sum  of  tlie  areas  of  the  curves  is 
the  area  of  the  S(juare.  Now  in  case  the  curve  is  such  that  the  frontiers  /„  and  l[^ 
formed  for  the  two  curves  are  not  zero,  it  is  clear  that  the  sum  Li  +  /-■  U>r  the 
tw(j  curves  will  not  give  the  area  of  the  square  but  a  smaller  area,  whereas  the 
sum  (Li  +  /„)  -\-  (L'l  4-  /,',)  will  give  a  greater  area.  Moreover  in  this  casi',  it  is  not 
easy  to  formulate  a  general  tlefinition  of  area  applicable  to  each  of  the  regions  and 
such  that  the  sum  of  the  areas  shall  be  eijual  to  the  area  of  the  combined  region. 
Hut  if   /„  ami  /^,  both  \anish,  then  the  sum  L,  -\-  L-  does  give  the  combined  area. 


ox  SI.MPLE  IXTEGEALS  313 

It  is  therefore  customary  to  restrict  the  application  of  the  term  ''area"  to  such  simple 
closed  curves  as  have  /„  =  0,  and  to  say  that  the  quadrature  of  such  curves  is  possible, 
but  that  the  quadrature  of  curves  for  which  l^^  ^pt  0  is  impossible. 

It  may  be  proved  that :  If  a  curve  is  rectlfiable  or  even  If  one  of  the  functions  x  (t). 
or  y{t)  is  of  limited  variation,  the  limit  /„  is  zero  and  the  quadrature  of  the  curve  is 
possUAe.  For  let  the  interval  t^'^t  -^t^  be  divided  into  intervals  A^i,  A.,i,  •  •  •  in 
which  the  oscillations  of  x  and  y  are  Cj,  e.,,  •  •  ■  ,  t;^,  ijo,  •  ■  •  •  Then  the  portion  of 
the  curve  due  to  the  interval  A,(  may  be  inscribed  in  a  rectangle  ejij,-,  and  that 
portion  of  the  curve  will  lie  wholly  within  a  rectangle  2ei-'2.rii  concentric  with 
this  one.  In  this  way  may  be  obtained  a  set  of  rectangles  which  entirely  contain 
the  curve.  The  total  area  of  these  rectangles  must  exceed  Z„.  For  if  all  the  sides 
of  all  the  rectangles  be  produced  so  as  to  rule  the  plane,  the  rectangles  which  go 
to  make  up  Au  for  this  ruling  must  be  contained  within  the  original  rectangles, 
and  as  Au>lu,  the  total  area  of  the  original  rectangles  is  greater  than  /„.  Next 
suppose  x{t)  is  of  limited  variation  and  is  written  as  J^  +  IT  {t)  —  X{t),  the  differ- 
ence of  two  nondecreasing  functions.  Then  2e,-  ^  n(ij)  +  -V(<,),  that  is,  the  sum 
(if  the  oscillations  of  x  cannot  exceed  the  total  variation  of  x.  On  the  other  hand 
as  y{t)  is  continuous,  the  divisinns  A,-^  could  have  been  taken  .so  small  that  r;,  <  t]. 
Hence 

/„  <  .1,,  ^  ^  2  e,-  .  2  t;,-  <  4  r;  ^  e,  S  4  7,  [U{t^)  +  X{t^)]. 

The  quantity  may  bo  made  as  small  as  desired,  since  it  is  the  product  of  a  finite 
quantity  by  rj.    Hence  /„  =  0  and  the  quadrature  is  i^ossible. 

It  may  be  observed  that  if  x  (t)  or  y  (t)  or  both  are  of  limited  variation,  one  or 
all  of  the  three  curvilinear  integrals 


—  jydx,         J-cdy,         l  jxdy  —  ydx 


may  be  defined,  and  that  it  should  be  expected  that  in  this  case  the  value  of  the 
integral  or  integrals  would  give  the  area  of  the  curve.  In  fact  if  one  desired  to 
deal  only  with  rectlfiable  curves,  it  would  be  possible  to  take  one  or  all  of  these 
integrals  as  the  definition  of  area,  and  thus  to  obviate  the  di.scussions  of  the  pres- 
ent article.  It  seems,  however,  advisable  at  least  to  point  out  the  problem  of 
quadrature  in  all  its  generality,  especially  as  the  treatment  of  the  problem  is  very 
similar  to  that  usually  adopted  for  double  integrals  (§  132).  From  the  present 
viewpoint,  therefore,  it  would  be  a  proposition  for  demonstration  that  the  curvi- 
linear integrals  in  the  cases  where  they  are  applicable  do  give  the  value  of  the 
area  as  here  defined,  but  the  demonstration  will  not  be  imdertaken. 

EXERCISES 

1.  For  the  continuous  curve  (27)  prove  the  following  properties: 

(a)  Lines  x  =  a,  x  =  h  may  be  drawn  such  that  the  curve  lies  entirely  between 
them,  has  at  least  one  point  on  each  line,  and  cuts  every  line  x  =  f ,  o<  f  <  6,  in  at 
least  one  point  ;  similarly  for  y. 

((3)  From  p  =  x  cos  a  +  y  sin  a.  the  normal  equation  of  a  line,  prove  the  prop- 
ositions like  those  of  (a)  for  lines  parallel  to  any  direction. 

(7)  If  (^.  77)  is  any  point  of  the  jy-plane,  show  that  the  distance  of  (^,  77)  from 
the  curve  has  a  minimum  and  a  maximum  value. 


314  l^TKGKAL  CALCULUS 

(5)  If  »t(|,  r)}  and  J/(f,  rj)  are  the  ininiimun  and  maximum  distances  of  (f,  7;) 
from  the  curve,  the  functions  7/1  (|,  77)  and  3/(|,  77)  are  continuous  functions  of  (f,  77). 
Are  tlie  coordinates  x{^.  77),  y(^.  rj)  of  the  points  on  the  curve  which  are  at  mini- 
nuim  (or  niaxinuim)  distance  from  (^,  77)  continuous  functions  of  (^,  77)  ? 

( e )  If  i',  i",  •  •  • ,  ^(^'^  •  •  •  are  an  infinite  set  of  values  of  t  in  the  interval  i^  =  i  ^  i^ 
and  if  i"  is  a  point  of  condensation  of  the  set,  then  x^  =  4>{t'^),  y'^  —  ^p  {V)  is  a  point 
of  condensation  of  tlie  set  of  points  (/',  ?/'),  (j",  2/"),  •••,  (j('->,  y^-^^),  •■■  corre- 
sponding to  the  set  of  values  t\  t"  •  •  ■ ,  U^\  •  •  • . 

(f)  Conversely  to  (e)  show  that  if  (x',  y'),  (x",  y"),  •  •  ■,  (j(*'),  y^'^^),  •  •  ■  are  an 
infinite  set  of  points  on  the  curve  and  have  a  point  of  condensation  (jc",  y^),  then 
the  point  (x",  y^)  is  also  on  the  curve. 

(77)  From  (f)  show  that  if  a  line  x  =  f  cuts  the  curve  in  a  set  of  points  y'.  y",  •  •  • , 
then  this  suite  of  y'a  contains  its  upper  and  lower  frontiers  and  has  a  maximum  or 
mininmm. 

2.  Define  and  discuss  rectifiable  curves  in  space. 

3.  Are  y  =  x-  sin  -  and  y  =  Vx  sin  -  rectitiable  between  x  =  0,  x  =  1  ? 

X  X 

4.  If  x{t)  in  (27)  is  of  total  variation  n  (^j)  +  X(ii),  show  that 

r'''P(x,  ./)cZx<J/[II(ii)  +  N(ii)], 

C  'J  ,'',1 

where  "SI  is  the  maximum  value  of  P(x,  //)  on  the  curve. 

5.  Consider  the  function  (9(^.  77,  /)  =  tan-i ^— -  which  is  the  inclination  of 

I  -  X  it) 

the  line  JDining  a  ]i<iint  (^,  77)  not  on  the  curve  to  a  point  (x,  //)  on  the  curve.    With 
the  notations  of  Kx.  1  (5)  show  that 

2  V5 

!A,^l  =  |6'(t,  77,  ^  + Ao-^(4. 7?,  0I< ttt;' 

m  —  2  J/3 

where  0  >  ■  Ax  |  and  5  >  |  A//  ]  \\\\\\  l>e  made  as  small  as  desired  by  taking  Af  sutiiciently 
small  and  where  it  is  assumed  that  m  ^  0. 

6.  From  Kx.  5  infer  that  ^(f.  77.  t)  is  of  limited  variation  when  t  describes  the 
ititerval  ^^  =  <  ^  ij  defining  the  curve.  Show  that  ^(|,  77,  t)  is  continuous  in  (f.  77) 
through  any  region  for  which  m  >  0. 

7.  Let  the  parameter  t  vary  from  ^,|  to  t^  and  suppose  tlie  curve  (27)  is  closed  so 
that  (x,  //)  returns  to  its  initial  value.  Show  that  the  initial  and  final  values  of 
^(^,  77,  i)  differ  by  an  integral  nuiltiple  of  2  tt.  Hence  infer  that  this  difference  is 
constant  over  any  region  for  which  )n  >  (t.  In  paitii'idar  show  that  the  constant  is 
0  over  all  distant  regions  of  the  plane.  It  may  he  remarked  that,  by  the  study  of 
this  change  of  0  as  i  describes  the  curve,  a  jiroof  may  lie  given  of  the  theorem  that 
the  chised  continuous  curve  divides  the  jilane  into  two  regions,  one  interior,  one 
exterior. 

8.  Extend  the  last  theorem  of  §  12o  to  rectifiable  curves. 


CHAPTER  XII 


ON  MULTIPLE   INTEGRALS 


129.  Double  sums  and  double  integrals.  Suppose  that  a  Iwdy  of 
matter  is  so  tliin  and  tiat  that  it  can  he  consichn-ed  to  lie  in  a  plane. 
If  any  small  portion  of  the  Ixxly  surrounding  a  given  point  J*(.r.  //)  l)e 
considered,  and  if  its  mass  be  denoted  hy  \iii  and  its  area  hy  A.l.  the 
average  (surface)  density  of  the  portion  is  the  quotient  A/// /A.l,  and  the 
actual  density  at  the  point  P  is  defined  as  the  limit  of  this  quotient 
Avhen  A- 1  =  0,  that  is,  . 

D(j;i/)  =  Yun  — -. 

The  density  may  vary  from  point  to  point.    Xow  conversely  suppose 

tliat  the  density  D(:r,  >/)  of  the  body  is  a  known  function  of  (r,  ]j)  and 

that  it  l)e  required  to  find  the  total  mass  of  the 

body.    Let  the  l)ody  l)e  considered  as  divided 

up  into  a  large  number  of  }»ieces  each  of  which 

is  )<ui<il!  1)1  ci-frij  (lu'i'rfinn.  and  let  A.l,-  be  the 

area  of  any  piece.     If  i^,-.  77,)  be  any  point  in 

A.L-,  the  density  at  that  })oint  is  l>{^i,  -q-)  and 

the  amount  of  matter  in  the  piece  is  a})proxi- 

mately  T)(^j.  77/) A.l,  provided  the  density  be  regarded  as  continuous, 

that  is,  as  not  varying  much  over  so  small  an  area.    Then  the  sum 

D(i^.  7?;)  A.  I J  +  ]>[t,  7/jA.l^+  ••■  +  />(t-„,  77,,;)  A.l,,  =^ />(>-,■,  77,-)  A.l,, 

extended  over  all  the  pieces,  is  an  ap])i'oximation  to  the  total  mass, 
and  may  be  suthcient  for  pi'actical  purposes  if  the  })ieces  l)e  taken 
tolej-ably  small. 

The  ]irocess  of  dividing  a  body  up  into  a  large  ninul)er  of  small  pieces 
of  which  it  is  regai'ded  as  the  sum  is  a  device  often  resorted  to  :  foi-  the 
]ir()}ierties  of  the  small  pieces  may  l)e  known  ap})roxiniately.  so  that 
the  corresponding  property  for  the  whole  body  can  l)e  ol)tained  apprcjx- 
imately  by  summation.  Tlius  by  detinition  the  moment  of  inertia  of  a 
small  ]iarticle  of  matter  relative  to  an  axis  is  //'/•-,  where  m  is  the  mass 
of  the  jiarticle  and  /■  its  distance  from  the  axis.  If  therefore  the 
moment  of  inertia  of  a  plane  body  with  res})ect  to  an  axis  perpendicular 

.315 


316  INTEGRAL  CALCULUS 

to  its  plane  were  required,  the  body  Avould  be  divided  into  a  large 
number  of  small  portions  as  above.  The  mass  of  each  portion  would 
be  approximately  J)($f,  7],^A.A^  and  the  distance  of  the  portion  from 
the  axis  might  be  considered  as  approximately  the  distance  r,-  from 
the  point  where  the  axis  cut  the  plane  to  the  point  (i^,  rj>)  in  the  por- 
tion.   The  moment  of  inertia  would  be 

or  nearly  this,  where  the  sum  is  extended  over  all  the  pieces. 

These  sums  may  be  called  double  sums  because  they  extend  over  tAvo 
dimensions.  To  })ass  from  the  approximate  to  the  actual  values  of  tlie 
mass  or  moment  of  inertia  or  whatever  else  might  be  desired,  tlie 
underlying  idea  of  a  division  into  parts  and  a  sul)sequent  summation 
is  kept,  but  there  is  added  to  this  the  idea  of  passing  to  a  lindt.  Com- 
pare §§16-17.    Thus 

Avould  l)e  taken  as  the  total  mass  or  inertia,  where  the  sum  over  n 
divisions  is  replaced  by  the  limit  of  that  sum  as  the  number  of 
divisions  becomes  infinite  and  each  becomes  small  in  every  direction. 
The  limits  are  indicated  by  a  sign  of  integration,  as 

lim  2^  /;  (^,,  7?;) A.  1 ,  =   Cn  (./■,  7/)  (lA ,        lim  '^  D  (^,,  77,)  r  ?A  J ,  =   f  DrHA . 

The  use  of  the  limit  is  of  course  dependent  on  the  fact  that  the  limit 
is  actually  approached,  and  for  practical  purposes  it  is  further  depend- 
ent on  the  invention  of  some  way  of  evaluating  the  limit.  ]-)oth  these 
questions  have  been  treated  when  the  sum  is  a  simple  sum  (§§  16-17, 
28-30,  35) ;  they  must  now  be  treated  for  the  case  of  a  double  sum  like 
those  above. 

130.  Consider  again  the  problem  of  finding  the  mass  and  let  7)^-  be 
used  briefly  for  7>(f,-,  t^,).  Let  M-  be  the  maxinmm  value  of  the  density 
in  the  piece  A.lj  and  let  vi;  be  tlie  mininTum  value.    Then 

7/^A.l,.s  7).A.1,.^.1/,.A/1;. 

In  this  Avay  any  approximate  expression  />,A.l,-  for  the  mass  is  shut  in 
between  two  values,  of  which  one  is  surely  not  greater  than  the  true 
mass  and  the  other  sui'elv  not  less.    Form  tlu>  sums 


^  »^.A.!.^^  />.A.l.^y  .1/,-A.l,; 


S 


extended  over  all  the  elements  A.4,-.    Noav  if  the  sums  .s-  and  N  approach 
the  same  limit  when  A.1,  =  0,  the  sum  27>,A.l.  which   is  constantly 


ox   IVIULTIPLE   INTEGRALS 


31^ 


inchuled  between  .s  and  .S'  must  also  ap})i'oa('li  that  limit  independently 
of  how  the  points  (if,  rji)  are  chosen  in  the  ai'eas  A.l,-. 

That  s  and  .S'  do  ap})i'oacli  a  common  limit  in  the  usual  case  of  a 
continuous  function  D(.r,  //)  may  be  shown  strikingly  if  the  surface 
s:  —  1)  (,r,  y^    be   drawn.     The 


term  />,A/lj-  is  then  repre- 
sented by  the  volume  of  a 
small  cylinder  upon  the  base 
A,l,-  and  with  an  altitude  equal 
to  the  height  of  the  surface 
.-;  =  1)  Qr,  v/)  above  some  point 
of  AJ,..  The  sum  2/>,A-',-  of 
all  these  cylinders  will  l)e  a})- 
proximately  the  volume  under 
the  surface  ,'w  =/>(,/■,  v/)  and 
over  the  total  area  .1  =  2A.I  •. 
The  term  M^AA  ■  is  re})resented 
by  the  volume  of  a  small  cylin- 
der u])on  the  base  A.l,-  and  cir- 
cumscri])ed  about  the  sui-face  ; 
the  term  ?;;;A.l,-,  by  a  cylinder 
inscribed  in  the  surface.  When  the  numbei-  of  chMucnts  A.I,-  is  increased 
without  limit  so  that  eacli  becomes  indctinitcly  small,  tlic  three  sums  s, 
S,  and  2/>,A.l,-  all  a})proach  as  their  limit  the  volume  under  the  surface 
and  over  the  area  .1.  Thus  the  notion  of  volume  does  for  the  double 
sum  the  same  service  as  the  notion  of  area  for  a  sim})le  sum. 

Let  the  notion  of  the  inteirral  be  applied  to  lind  tlie  formula  for  the  center  of 
graxity  of  a  plane  lamina.    .V.ssuine  that  the  rectaiifruhir  coordinates  of  tlie  center 
of  gravity  are  {jr.  y).    ('(insi(k'r  the  bedy  as  divi(U»d  into  small  areas  A^,.    If  (^,-,  ?/,•) 
is  any  point  in  the  area  AJ,-.  the  approximate  moment  of 
the  approximate  mass  J>»,A.l,-  in  that  area  with  respect  to 
the  line  x  =  x  is  the  product  (^,-  —  x)Z),A^l,-  of  the  mass 
by  its  distance  from  the  line.    The  total  exact  moment 
would  tlierefore  be 

limV  {^i  -  l)]J,AAi  =  C{x  -  x)IJ{x,  y)clA  =  0, 

and  uuist  vanish  if  the  center  of  gravity  lies  on  the  line 
X  =  X  as  assumed.    Then 

fxD  (x,  7/)  (lA  -  CxB  (x,  y)  dA  z^  0     or    CxDdA  =x  Cl)  (x,  y)  dA . 

These  formal  operations  presuppose  the  facts  that  the  difference  of  two  integrals  is 
the  intei^ral  of  the  difference  and  that  the  integral  of  a  constant  x  times  a  functi<jn  B 


.318 


INTEGRAL  CALCULUS 


is  till'  product  of  tlie  constant  by  the  iutcu'i'al  of  the  function.  It  should  be  imme- 
diiitoh'  apparent  that  as  these  rules  are  applicable  to  sums,  they  must  be  applicable 
to  the  limits  of  the  sums.   The  ecpiatiun  may  now  be  solved  for  x.   Then 


CxDdA        fxdni  fylXlA        C  ydm 


f 


UdA 


f 


UdA 


(1) 


where  m  stands  for  the  mass  of  the  body  and  dm  for  UdA,  just  as  A//*,  miirht  replace 
UjAAj ;  the  result  for  y  may  be  written  down  from  symmetry. 

As  another  example  let  the  kinetic  energy  of  a  lamina  moving  In  Ita  plane  be  cal- 
culated.   The  use  of  vectors  is  advantaiieous.    Let  ro  be  the 
vector  from  a  lixed  oriuin  to  a  point  which  is  fixed  in  the 
body,  and  let  ri  be  the  vector  from  this  point  to  any  other 
point  of  the  b(xly  so  that 

dti        dTr,        dli ; 

r,  =  ro  +  ri ,-,         ~rr  = -rr  +  —,r    ^^'    V'  =  "^o  +  '^i' • 
dt        dt         dt 

The  kinetic  ener;.;y  is  2  I  vjAiitj  or  better  the  inte::ral  of  I  v-dm.    Now 

VJ  =  V/.V;  =-  V,,.V,j  +  Vi,-.Vi,-  +  2v,j.Vi,-  =  r~  +  /Y,a)-  4-  2Vm.Vi,-. 

Tiiat  Vi,.Vi,  =  /'fi-oj-.  where  ;-i,-  =  |ri,-'  and  w  is  the  an,f;-nlar  velocity  of  the  Itody 
about  tlie  point  r,,.  follows  from  the  fact  that  ri,-  is  a  vector  of  constant  length  ri,- 
and  hence  dr^  =  r-ndO.  where  dd  is  the  angle  that  rn  turns  through,  and  conse- 
(juently  ai  =  d6/dl.    Next  integrate  i>ver  the  body. 

I  \  t-dm  =  I  },  vijdiii  +  j  i  r'lw-dm  +   i  v.j.Vidm 

=  }.  vijM  +  io)-  /  r{dm  +  v,j'  (  Vidm  ;  (2) 

for  r,j-  and  u-  are  constants  relative  to  the  integration  over  the  body.    Note  that 

v„.  I  v,'///(  =  0     if     Vo  =  0     or  if     fy^d)n  =  j   '    r^din  =  -     fi^ilm  =  0. 

15ut  V,,  =  0  holds  only  when  the  ixiiut  r,,  is  at  rest,  and  I  r^d)n  =  0  is  the  condition 
that  r,  be  the  center  of  L:ravit\'.    Jn  the  last  case 


T=    Cl  r-,hn  r^  \  i\;}f  + 


I 


/■V' 


Im. 


As  /  is  the  integral  whirh  lias  been  called  the  nioiiient  of  inertia  relative  to  an  axis 
through  the  jioiiit  r,,  per]Mnilicnlar  to  the  plane  of  the  boily.  the  kinetic  eneru-y  is 
seen  to  be  the  sum  of  '  Mr-,  whidi  Would  be  the  kinetic  ener:^y  if  all  the  mass  were 
concentrated  at  the  iciitn-  ..f  gravity,  ami  of  '  Iw'-.  wliicli  i>  the  kinetic  energy  of 
rotation  about  the  ceiitei-  oi'  -ravity  :  in  ca>e  r,,  imlicated  a  jioint  at  lest  (even  if 
only  instantaiU'ousl_\'  as  in  ^  '■','■  i)  the  whole  kinetic  energy  would  reduce  to  the 
kinetic  energy  of  rottition  I  Iw-.  In  case  r,,  indicated  neither  the  center  (jf  gravity 
nor  a  point  at  rest,  the  third  tei-m  in  (li)  would  not  vanish  and  the  expre.-sion  for 
the  kinetic  energy  woiUd  be  more  complicated  owinu'  to  tlie  preseiire  of  this  term. 


( )N   31 U  LTIPLE   IN  TE( T K  A L8 


319 


SCi    X 


131.    To  eriiliijifc  tJic  doiihli'  infc(ii'<il  In  cisc  flw  rfr/ian  is  ii  rcr'ton'/Zi' 

pdrallel  to  tlie  (/.res  of  cooriUnnti's,  let  the  division  be  made  into  small 

rectangles    by  drawing   lines    ])arallel    to   the 

axes.    Let  there  he  in  equal  divisions  on  one      ^^] »?  columns /=i, 2..., ;n 

side  and  n  on  the  other.    There  will  then  l)e  j^^f.i'.zij'-V-. 

II)  n  small  pieces.    It  will  Ije  convenient  to  in-         '''^■^n'-irt;:::': 

troduce  a  double  index  and  denote  by  AJ,--  the      ,,         -"■:t:'"-t: 

•'  uv [  '  '  '  • 

area  of  the  rectangle  in  the  /th  colunni  and/th      -(j. — ^ 

row.    Let  (^,-y,  ?;,/)   be  any   ])oint.  say  tlie  mid- 
dle point  in  the  area  A.l,-  =  A./--A//;.    Then  the  sum  may  l)e  written 

^  D(i^j,  77,;)  A. I,^  =  /^nAr^A//,  -f  7>,,A.''-.A//i  +  •  •  •  +  J>,.,A'';,A'/i 
'•■'■  -h  7-'i,A,/'iA//._,  +  ]).^^\.r.\ii.,  H +  7^„,,A.'-,„A//. 

+ 

-f  /)i„A,''iA.v,.  -f  7>o„A-''-..^//„  H h  J>,„„X''„A!/,r 

Xow  the  terms  in  the  tirst  row  are  the  sum  of  the  contriljutions  to 
1;j  of  the  rectangles  in  the  first  row,  and  so  on.     Hut 


and 


A//,-^    Pi^;.    7;;)A./',-    = 


JM.r,  r].)>/.r  +  ^, 


A/j- 


That  is  to  say.  by  taking  in  sufhcicntly  lai'ge  so  that  the  individual 
increments  A.'-,-  ai'c  sufHciently  small,  tlie  sum  can  lie  made  to  differ 
from  the  integral  by  as  little  as  desired  lu'cause  tlie  integral  is  l)y 
definition  the  limit  of  the  sum.    In  fact 


'lA^^\M,-in, 


A./ 


i  =  t  <  ■ 


if  e  lie  the  maximum  variation  of  D(.i\  ;,)  over  one  of  the  little  rectangles. 
Aftei'  thus  suniming  up  according  to  rows,  sum  u]i  the  I'ows.    Then 

^7>,;A-I,,.  =   /     'yyt./',  7?;)'/,'-AVj  +  /     ']>[■'■.  T^jd.rX,/,, 

+  ■■■  +  j      '}>(■'■■  r;„i'/.''A,V„  +  A, 
[Aj  =  [lA.'^i  +  ^-j-^/Z-j  +"  •  ■  ■  +iA.'/,.    =  ei'''  —  •'•,i'V  A//  =  £(.'•  —  ,'•„)(//  —  .y„) 
If  /      J>(.'\  //)'/,'•  =  <^(.'/). 

then         2}  7>,yA.  I ,-;  =  (^  ( -,;, )  A//^  -b  c^  ( 7;., )  A //.,  -J h  <^  ( 7/,, )  A //„  +  A 

=  /      4>[/J)'^l/  +  K  +  X,  K,  A  small. 


320 


IXTECIKAL   CALCULUS 


Hence  * 


(3) 


It  is  seen  that  the  douV)le  inteij^ral  is  equal  to  the  result  obtained  by 
tirst  integrating  with  res})eet  to  ./■,  regarding  //  as  a  parameter,  and  then, 
after  substituting  the  limits,  integrating  with  respect  to  y.  If  the  sum- 
mation had  l)een  first  according  to  columns  and  second  according  to 
rows,  then  by  symmetry 


JDdA  =  J     T  '/^(.z-,  !i)dn];,  =j    j     Vj(,v,  il)'bjdx. 


(3') 


This  is  really  nothing  l>ut  an  integration  under  the 
sign  (§  120). 

//'  tlie  rrrjion  orcr  jrjilrli  ihc  siinnnntlon  is  twf ended 
i.s  not  a  recf(in(ili'  jKirallel  fa  the  axes,  the  method 
could  still  be  apjilied.  IJut  after  summing  or  rather 
integrating  according  to  rows,  the  limits  woidd  not 
l)e  constants  as  x^  and  x^,  but  would  be  those  func- 
tions X  =  <^o(//)  mid  ■'■  =  </)///)  of  //  which  rei)resent  the  left-hand  and 
ri'dit-hand  curves  which  bound  the  req-ion.    Thus 


]jdA=  (  D(x,y)dxdy. 

And  if  the  summation  or  integi'ation  had  l)een  iirst 
with  res})ect  to  columns,  the  limits  would  not  have 
been  the  constants  //,.  and  //^  but  tlic  functions 
ij  =  ipjx)  and  //  =  i/Zjl.'')  which  rciircscnt  the  lowci- 
and  up})er  bounding  curves  of  the  I'cgion.     Tlius 


(3") 


J"'"=£JZ"'"' 


•,   I/)  dijdx. 


dx  a'l  A' 


<•'  ; 


The  order  of  the  integrations  cannot  be  inverted  witlmnt  making  tlic 
corresponding  changes  in  tlie  limits,  the  tii-st  set  of  limits  being  sucli 
functions  (of  tlie  \'ariable  witli  I'cgard  to  whieli  tlie  second  integration  is 
to  be  performed )  as  to  sum  u]»  according  to  strips  reaching  from  (jiie  side 
of  the  region  to  tlie  otlier,  and  tlie  second  set  of  limits  being  constants 
which  determine  the  extreme  limits  of  tlie  second  variable  so  as  to  sum 
up  all  the  strips.  Althougli  the  results  (3")  and  (.'V")  are  equal,  it  fre- 
(juently  happens  tliat  one  of  them  is  decidedly  easier  to  evaluate  than  the 
other.    ^Moreover,  it  has  clearly  been  assumed  that  a  line  }iarallel  to  the 


The  result  niav  al 


ililaiiii'il  as  ill  ]v\.  S 


ox  ^MULTIPLE  INTEGRALS  321 

axis  of  tlie  first  integration  cuts  the  bounding  curve  in  only  two  points  ; 
if  this  condition  is  not  fulfilled,  the  area  must  be  divided  into  subareas 
for  which  it  is  fulfilled,  and  the  results  of  integrating  over  these  smaller 
areas  must  l)e  added  algel)raically  to  find  the  complete  value. 

To  apply  tliese  rules  for  evaluating  a  double  integral,  consider  the  prf)l)leni  nf 
finding  the  moment  of  inertia  of  a  rectangle  of  constant  density  with  respect  to 
one  vertex.    Here 

7  =  jih-MA  =  i)j{x-^  +  y"-)aA  =  dJ  Y"  (•='  +  y^-)'^^^'iy 

=  J)f  T '.  /^  +  ///2  I " (1;/  =:  ])  f  \i  a'  +  «2/2) chj  =  1  Lnb (a^  +  I;^). 

J        \_  Jo  J 'I 

If  the  problem  had  been  to  find  the  moment  of  inertia  of  an  ellipse  of  uniform 
density  with  respect  to  the  center,  then 

//• ''     /"•  +  {■  V  ft-  — '/'- 
(/-  +  //•-)  '1A=1>  I    „"   {x"~  +  7/2)  dxdy 


1) 


0        ■     , 
J  -  a     J  -  -  \  II-  —  .1- 


Either  of  these  forms  might  be  evaluated,  but  the  moment  of  inertia  of  the  whole 
ellipse  is  clearly  four  times  that  of  a  quadrant,  and  hence  the  simpler  results 


{x-  +  y-)  dxdy 


{X-  +  y-)'hjdx  =  -  Ihih  (r/.2  4-  Jf). 
4 

It  is  highly  advisal)le  to  make  use  of  symmetry,  wherever  possible,  to  reduce  the 
region  over  which  the  integration  is  extended. 

132.  With  regard  to  the  more  careful  consideration  of  the  limits  involved  in  the 
definition  of  a  double  integral  a  few  observations  will  be  .sufHcient.  Consider  the 
sums  .S  and  -s  and  let  .U,A-1,-  lie  any  term  of  the  first  and  ?H,A.l,-  the  corresponding 
term  of  the  second.  Suppose  tlie  area  AJ,-  divided  into  two  parts  A^li,-  and  A^lo,-, 
and  let  .Vi,-.  J/j,-  Ije  the  maxima  in  the  i>arts  and  ??ii,-,  nia  the  minima.  Then  since 
the  maxinnun  in  the  whole  area  A.l,-  cainiot  be  le.ss  than  that  in  either  part,  and 
the  minimum  in  tlie  whole  cannot  lie  greater  than  that  in  either  part,  it  follows 
that  m\i  s  Hi,-,  m-^i  s  m,-,  J/j,-  ^  ^f,-,  ,Vo,-  ^  3/,-,  and 

mi^Ai  ^  »/i,A.li;  +  m.2;^A■2i,         3/i,A.ii,-  +  3/o/AJo/  ^  .V,A.l,-. 

Hence  when  one;  of  the  pieces  A^l,-  is  .subdivided  the  sum  .S'  cannot  increase  nor  the 
sum  s  decrease.    Then  contimied  inequalities  may  be  written  as 

mA  ^  Vm,AJ,-  ^  Vi'(i^•.  Vi)^-^i  ^  V-ViA.l,-  s  ^[A. 

If  then  the  original  divisions  A.l,-  be  subdivided  indefinitely,  both  .'^'  and  -s  will 
approach  limits  {§§  21-22)  :  and  if  those  limits  are  the  same,  the  simi  27^,A.l,-  will 
approach  that  common  limit  as  its  limit  independently  of  how  the  jxjints  (^,-,  iji) 
are  chosen  in  the  areas  A-1,-. 


322  INTEGKAL  CALCULUS 

It  lias  not  been  .sliowii,  however,  that  the  limits  of  S  and  s  are  independent  of 
the  method  of  division  and  subdivision  of  the  whole  area.  Consider  therefore  not 
only  the  sums  6'  and  s  due  to  some  particular  mode  of  subdivision,  but  consider  all 
such  sums  due  to  all  possible  modes  of  subdivision.  As  the  sums  S  are  limited 
below  by  niA  they  must  have  a  lower  frontier!,,  and  as  the  sums  s  are  limited 
above  by  MA  they  must  have  an  upiier  frontier  /.  It  must  l)e  shown  that  /  ^  i>. 
To  see  this  consider  any  pair  of  sums  .S  and  s  corresponding-  to  one  division  and 
any  other  pair  of  sums  S'  and  .s'  corri'spondinL;'  to  anotlier  method  of  division  ;  also 
the  sums  S"  and  .s"  (.-orrespondiim-  to  the  division  obtained  by  combining,  that  is, 
by  superposing  the  two  methods.    Now 

,S"  s  >■"  s  s"  s  .s,       ,S'  s  S"  s  .s"  ^  ,s',       ,s  s  L.       S'  s  L,       ,s  ^  /,       s'  s  /. 

It  therefore  is  seen  that  any  »S  is  greater  than  any  .s.  wjiether  these  sums  correspond 
to  the  same  or  to  different  methods  of  subdivision.  Now  if  L<1.  some  .S'  would 
have  to  be  le.ss  than  some  .s  ;  for  as  I.  is  the  frontier  for  the  sums  .s,  there  nuist  lie 
some  such  siuns  which  differ  by  as  litth.'  as  desired  from  L  ;  and  in  like  manner 
tln're  must  be  some  sums  s  which  differ  by  as  little  as  desired  from  I.  Hence  as  no 
^'  can  be  less  than  any  s,  the  supposition  i  <  Hs  untrue  and  L  ^  I. 
Now  if  for  any  method  of  division  tlie  limit  of  the  difference 

lim  (N  -  .n)  =  lim  V  (.V,-  -  ??(,-)  A.l,-  =  lini  V  0,A.l.-  =  0 

of  the  two  stims  corresponding  to  that  method  is  zero,  tlie  frontiers  L  and  /  nuist  be 
tlie  same  and  both  N  and  .s  approach  tliat  cf>nnnon  value  as  their  limit;  and  if  tlie 
difference  S—s  approaches  zero  for  ever\'  method  of  division,  tlie  sums  S  ami 
.s  will  approach  the  same  limit  L  =  /  for  all  methods  of  division,  and  the  sum 
-D;AAi  will  approach  that  limit  iiiilepeinleiitly  of  the  method  of  division  as  well 
as  independently  of  the  selection  of  (f,-.  t?/).  Tliis  result  follows  from  the  fact  that 
L  —  /  ^  ,S  —  ,s,  <S  —  L  ^  S  — .S-.  /  —  -s-  ^  N  —  .N,  and  hence  if  the  limit  of  S  —  .s  is 
zero,  then  L  =  l  and  N  and  .s  must  a]iproacli  the  limit  L  =  I.  One  case,  which 
covers  those  arising  in  iiractice.  in  which  these  residts  are  true  is  that  in  which 
7>(.r.  //)  is  contiiuious  over  the  area  A  except  i)erhaps  upon  a  linite  number  of 
curves,  each  of  which  maybe  iiicloscd  in  a  strip  of  area  as  .'^mall  as  desired  and 
upon  which  J){.r.  //)  remains  tinite  though  it  be  discontiiuious.  Tor  let  the  curves 
over  wliich  D(x.  y)  is  discontinuous  be  inclosed  in  strii)s  of  total  area  a.  'J'he  cou- 
trihutioii  of  these  areas  to  the  difference  N  —  .s  cannot  exceecl  (M  —  ni)(i.  .\]iart 
fi-oiii  these  areas,  the  function  J)(r.  //)  is  continuous,  and  it  is  ]>ossilile  to  take  tlie 
divisions  A.l,-  so  small  that  the  OM'iUaiion  uf  the  function  over  any  one  of  them 
is  less  tlian  an  assiiineil  numbiT  e.  Hence  the  contribution  to  N  —  .s  is  less  than 
6(.l  —  ")  for  tlie  remaining  uudeh'tiMl  regions.  The  total  value  of  N  —  .s  is  tlierc- 
foi'c  less  than  (3/—  iii)ii  +  e(.l  —  u)  ami  can  certainl}'  be  made  as  small  as  desired. 
Tlie  proo^  of  the  existence  and  uiii(iueness  of  the  limit  of  ^DjAA,-  is  therefore 
obtained  in  case  I)  is  continuous  over  the  region  .1  except  for  i)oiiits  along  a  finite 
lutmber  of  cttrves  ^\•here  it  may  be  discontinuous  provided  it  I'emains  finite. 
Throughout  the  discussion  tlie  term  "  area  ""  has  been  applied  :  this  is  justified  by  the 
previous  woriv  (§]2S).  Instead  of  di\i(ling  the  area  A  into  elements  A.I.  one  may 
rule  the  area  with  lines  jmrallel  to  the  axes,  as  done  in  ij  12t<.  ami  consider  the  sums 
^MAxAi/.  ZmA.rA;/.  -hA.rA;/.  where  the  first  sum  is  exteinleil  over  all  the  reclaii- 
uh's  wliich  lie  within  or  upon  ihe  cui'\c.  where  the  second  sum  is  extended  o\(r 
all  the  rectanu-les  within  the  curvt'.  and  where  the  last  extends  over  all  rectaimlcs 


ON  MULTIPLE  INTEGRALS  323 

within  the  curve  and  over  an  arbitrary  number  (if  tliose  upon  it.  In  a  crrtain 
sense  tliis  metliod  is  simpler,  in  that  tlie  area  tlien  falls  out  as  the  intei;ial  of  the 
special  function  which  reduces  to  1  within  the  curve  and  to  0  outside  the  curve, 
and  to  either  upon  the  curve.  The  reader  who  desires  to  follow  this  method  throiiiih 
may  do  .so  for  himself.  It  is  not  within  the  range  of  this  book  to  do  more  in  the 
way  of  rigorous  analysis  than  to  treat  the  simpler  questions  and  to  indicate  the 
need  of  corresponding  treatment  for  other  questions. 

The  justification  for  the  method  of  evaluating  a  definite  double  integral  as  given 
above  offers  some  difliculties  in  case  the  function  X>(j,  y)  is  discontinuous.  The 
proof  of  the  ride  may  be  obtained  liy  a  careful  con.sideration  of  the  integration  of 
a  function  defined  by  an  integral  containing  a  parameter.    Consider 

cpiij)  =/''^(-'-.  .'/)'?.'•,         f"''P{l/)dy  =f"'f'''lJ{.c,  y)d.aly.  (4) 

It  was  seen  (§  118)  that  (piy)  is  a  contiiuious  function  of  y  if  D{x,  y)  is  a  con- 
tinuous fiuiction  of  (x,  y).  Suppose  that  l>{.t,  y)  were  discontinnotts,  but  remained 
finite,  on  a  finite  number  of  curves  each  of  which  is  cut  by  a  line  parallel  to  the 
a^-axis  in  oidy  a  finite  munln'r  of  jxiints.  Form  A(p  as  liefore.  Cut  out  the  short 
intervals  in  which  discontinuities  may  occur.  As  the  number  of  such  intervals  is 
finite  and  as  each  can  be  taki'U  as  short  as  desired,  their  total  contribution  to  (p{y) 
or  (p{y  +  Ay)  can  be  made  as  small  as  desired.  For  the  remaining  portions  of  the 
interval  JC^,^  x  ^  d\  the  previous  reasoinng  applies.  Hence  the  difference  A<p  can 
still  be  made  as  small  as  desiivd  and  0  (y)  is  continvious.  If  D{x,  y)  be  discontinuous 
along  a  line  y  =  ^  paralU'l  to  the  ,r-axis.  then  <piij)  nnght  not  be  defined  and  nnght 
have  a  discontinuity  for  the  value  y  =  ji.  But  there  can  be  only  a  fiinte  num- 
ber of  such  values  if  U(x.  y)  satisfies  the  coiulitions  imposed  upon  it  in  considering 
the  double  integral  above.    Hence  cp  {y)  would  .still  be  integrable  from  y^^  to  y-^.  Hence 

/     '  /     'Z>(,r.  y)dxdy         exi.sts 
m  {x^  -  3-,,)  (//i  -  y,)  ^  j  '  j  'D{x,  y)  dxdy  ^  M{x^  -  x„)  (y^  -  ?y,j) 


and 


Add 

and  ^      ' 


under  the  conditions  imposed  for  the  doulile  integral. 

Now  let  the  rectangle  .c,,  =  ,/•  =  ,Cj.  y^  s  ;/  ^  y^  he  divided  up  as  before.    Then 

mi;Ax;Ayj  ^  I  '     |  l){x,  y)dxdy  ^  ^f;jAiXAJy. 

^  m;jAXiAyj  ^  ^  J  "  '  ^ ' "/  '  ^  ^' '  ^^  (■''•  !/)  <?■'■'?//  ^  ^  '^'''■'^'■''^J'-' 
V    r  ■"  ^  ■"■'"  f  '  ^  ^''  1>  (X.  y)  dxdy  =  f  ■"'  f'T)  {x.  y)  dxdy. 
Now  if  the  numVier  of  divisions  is  multiplied  indefinitely,  the  limit  is 

r"'  f'lH-c,  y)dxdy  =  lim  Vz/'z/A-l,;/  =  lim  V  .U,;/A.l//  =  fD(x.  y)dA. 

Thus  the  previous  rule  for  the  rectangle  is  proved  with  proper  allo\vance  for  pos- 
sible discontinuities.  In  case  the  area  .1  did  not  form  a  rectangle,  a  rectani:le 
could  be  described  aViout  it  and  the  fiuiction  l){x,  y)  could  be  deluied  foi-  the 
whole  rectangle  as  follows:  For  points  within  ^1  the  value  of  l>{x.  y)  i>  already 


\dA. 


324  INTEGRAL  CALCULUS 

defined,  for  points  of  the  rectangle  outside  of  A  take  I){x,  y)  =  0.  The  discon- 
tinuities across  the  boundary  of  A  wliicli  are  tlius  introduced  are  of  the  sort 
allowable  for  either  integral  in  (4),  and  the  integration  when  applied  to  the  rec- 
tangle would  then  clearly  give  merely  the  integral  over  A.  The  limits  could  then 
be  adjusted  so  that 

I      D  (x,  y)  dxdy  =1  D  i^-,  v)  dxdy  =  I  IJ  (x,  ?/)  ( 

The  rule  for  evaluating  the  double  integral  by  repeated  integration  is  therefore 
proved. 

EXERCISES 

1.  The  sum  of  the  moments  of  inertia  of  a  plane  lamina  about  two  perpendicular 
lines  in  its  plane  is  equal  to  the  moment  of  inertia  about  an  axis  j)erpendicular  to 
the  plane  and  passing  through  their  point  of  intersection. 

2.  The  moment  of  inertia  of  a  plane  lamina  about  any  point  is  equal  to  the  sum 
of  the  moment  of  inertia  about  the  center  of  gravity  and  the  product  of  the  total 
mass  by  the  square  of  the  distance  of  the  point  from  the  center  of  gravity. 

3.  If  \\\)on  every  line  issuing  from  a  point  0  of  a  lamina  there  is  laid  off  a  dis- 
tance OP  such  that  OP  is  inversely  proxjortional  to  the  square  root  of  the  moment  of 
inertia  of  the  lamina  about  the  line  OP,  the  locus  of  P  is  an  ellipse  with  center  at  0. 

4.  Find  the  moments  of  inertia  of  these  uniform  laminas: 
(a)  segment  of  a  circle  about  the  center  of  the  circle, 

(/3)  rectangle  about  the  center  and  about  either  side, 

(7)  parabolic  segment  bounded  by  the  latus  rectum  about  the  vertex  or  diameter, 

(5)  I'ight  triangle  about  the  right-angled  vertex  and  about  the  hypotenuse. 

5.  Find  by  double  integration  the  following  areas: 

(a)  quadrantal  segment  of  the  ellipse,         (/3)  between  y-  =  x^  and  y  =  x, 
(7)  between  S  y^  =  25 x  and  5x"  =  9  //, 
(5 )  between  x^  +  v/-  —  2 x  =  0,  x'^  +  y'^  —  2  y  =  0, 

(e)  between  y^  =  4ax  -|-  4a-,  ij-  =  —  Ahx  -{■  4//-^, 

(f )  within  {ij  -  X-  2)^  =  4  —  x^ 

(77)  between  x-  =  4  a//,  ?/(x'-^  +  4:tfi)  =  8  a''', 
{6)  y^  =  (IX,  X-  +  //-  —  2 «x  =  0. 

6.  Find  the  center  of  gravity  of  the  areas  in  Ex.  .5  (a),  (/3),  (7),  (5),  and 

222 
(a)   quadrant  of  a*y~  =  a'-x'*  —  x^,  (/3)  quadrant  ot  xs  +  ys  =  as, 

(7)  between  x2  =  y/a  +  ai,  x  +  v/  =  a,         (S)  segment  of  a  circle. 

7.  Find  tlui  volumes  under  the  surfaces  and  over  the  areas  given  : 


((r)  spliei'c  2  =  Va^  —  x-  —  //-  and  s(juare  inscribed  in  x-  +  //-  =  a^, 

{(3)  splicre  z  =  Va-  —  x-  —  //-  and  circle  x-  +  //-  —  ax  =  0, 

{7)  cylinder  z  =  V4a-  —  //-  and  circle  x-  +  //-  —  2  <ix  =  0, 

(5)  paraboloid  z  =  kxi/  and  rectangle'  0  ^  x  ^  a,  0  =s  v/  =  ^>, 

(e)  iiaraboloid  z  =  kxy  and  circle  x-  +  ?/-  —  2  ax  —  2  a//  =  0, 

(f )  plane  x/a  +  y/h  +  z/r^  =  1  and  triangle  x//  (x/a  +  y/^J  —  f )  =  0, 
(7/)  paraboloid  z  =  \  —  x~/~i  —  i/'^/d  above  the  ])lane  2  =  0, 

{6)  paraboloid  2  =  (x  +  y)-  and  circle  x-  +  i/~  =  a-. 


ON  MULTIPLE  INTEGRALS  325 

8.  Instead  of  choosing  (^,-,  rij)  as  particular  points,  namely  the  middle  points,  of 
the  rectangles  and  evaluating  Si>(|j-,  tjj)  AXiAyj  subject  to  orr(n-s  X,  k  which  vanish  in 
the  limit,  assume  the  function  -D(x,  y)  continuous  and  resolve  the  double  integral 
into  a  double  sum  by  repeated  use  of  the  Theorem  of  the  Mean,  as 

(p{i/)=  I     J){x.,  y)dx  =^]  D{^i,  !/)Ajri,         |'s  properly  chosen, 

J^ 'V ii/) dy  =  ^ -^ (ni) Mj  =X\X^^ ^'' '  ''■'■^ ^""'l ^■^■'  ^X'^ ^"■'  '^'^ ^^^''- 

9.  Consider  the  generalizatidu  of  Osgood's  Theorem  (§35)  to  apply  to  double 
integrals  and  sums,  namely:  If  ii'ij  are  inlinitesimals  such  that 

^0  =  l>{i;,  Vi)AA;j+  ^liAAri, 

where  t^/j  is  uniformly  an  iuiinitesimal,  then 

lim  2  "//  =  r^^A  y)dA  =  r"'  r''^{-^,  y)dxdy. 

Discuss  the  statement  and  the  result  in  detail  in  view  of  §  ;-54. 

10.  Mark  the  region  of  the  j"//-plane  over  which  the  integration  extends:* 

{a)    f'f  Ddydx,  (13)    f  f '"'  Ddydx,  (y)   f'  f  "  Iklxdy, 

Jo    Jo  J  I     J.r  Jo    J  ir 

(5)    /  r        Jkhjdx,    (e)    /    ^    I  l)dnl<p,     (f)  j        j    ^         -"Dd,pdr. 

'  — ^  ^  b        "  "  ~  b 

11.  The  density  of  a  rectangle  varies  as  the  square  of  the  distance  from  one 
vertex.  Find  the  monunit  of  inertia  about  that  vertex,  and  about  a  side  through 
the  vertex. 

12.  Find  the  mass  and  center  of  gravity  in  Ivx.  11. 

13.  Show  that  the  moments  of  momentum  (§80)  of  a  lamina  about  the  origin 
and  about  the  point  at  the  extremity  of  the  vector  r,,  .satisfy 

I  rxvdin  =  r^iX  /  vdm  +  |  r'xvtZm, 

or  the  difference  between  the  mdiiients  of  momentum  about  P  and  Q  is  the  moment 
about  P  of  the  total  momentum  considered  as  applied  at  Q. 

14.  Sliow  that  the  formulas  (1)  fur  the  center  of  gravity  reduce  to 

f  xyDdx  f   lyyDdx  (    ^  x{y.y  —  y^^Bdx 

-  Jo  -  Jo      '  -  J.r. 

x  = ,  1/  =       -         -        or     X  —  — ^ , 

I     i/lMx  (  !/l)dx  C  '  \i/i  —  yo)  Ddx 

■'() 

J     '  I  (.'/]  +  l/o)  il/i  -  i/o)  ^dx 


y  = 


f  \y^-y^)Bdx 


*  Exercises  involving  polar  coordinates  may  be  postponed  until  §  1,'U  is  reached,  unless 
the  student  is  alreatly  somewhat  familiar  witli  the  sul)jcct. 


'di/dx. 


32G  INTEGRAL  CALCULUS 

wlicn  D{x,  y)  reduces  to  a  function  J){x),  it  beiui,'  understood  that  for  the  first 
two  the  area  is  bounded  by  x  =  0.  x  =  «,  1/  =/(j-),  y  =  0,  and  for  the  second  two 
by  X  =  j-y,  X  =x^,  ?/i  =/i(x),  //,j  ^  ,/;,(.'•). 

15.  A  rectanguh^r  hole  is  cut  throuuli  a  spliere,  tlie  axis  of  the  hole  being  a 
diameter  of  the  spliere.  Find  the  volume  cut  out.  Discu.ss  the  problem  by  double 
inteuration  and  also  as  a  solid  with  parallel  ba.ses. 

16.  Show  that  the  moment  of  momentum  of  a  plane  lamina  about  a  fixed  point 
or  about  the  instantaneous  center  is  7w,  where  w  is  the  angular  velocity  and  7  the 
moment  of  inertia.  Is  this  true  for  the  center  of  gravity  (not  necessarily  fixed)? 
Is  it  true  for  other  points  of  the  lamina? 

18.  In  these  integrals  cut  down  the  region  over  which  the  integral  nmst  be 
extended  to  the  smallest  possible  by  tising  synnnetry,  and  evaluate  if  possible: 

{a)  the  integral  of  Ex.  17  with  1)  =  7/^  —  2x'-i/. 

(13)  the  integral  of  Ex.  17  with  JJ  =  (x  -  2  V';3)-y  or  7;  =  G  _  2  V.S)?/, 

(7)  the  integral  of  Ex.  10(e)  with  i*  =  r(l  +  cos0)  or  I)  =  sin  <p  cos  0. 

19.  The  curve  y  =f{x)  between  x  =  a  and  x  =  h  is  constantly  increasing. 
Express  the  volume  olitained  by  revolving  the  curve  about  the  x-axis  as 
7r[f  {(()]- {h  —  <i)  i)lus  a  double  integral,  in  rectangular  and  in  polar  coordinates. 

20.  Express  tlie  area  of  the  eardioid  r  =  a  (I  —  cos0)  by  means  of  double  inte- 
gration in  rectangular  coiirdinates  with  the  limits  for  both  orilers  of  integration. 

133.  Triple  integrals  and  change  of  variable.  In  tlic  (>xteiisioii  from 
(l()iil.)le  to  triple  iind  lii^'lier  integrals  there  is  little  to  cause  ditticiilty. 
For  the  dis(;iissioii  of  the  triph',  integral  the  saiiu^  fouiulation  of  mass 
and  density  may  be  made  fiindatnental.  If  D(-'',  //,  '-')  is  the  density  of 
a  l)odv  at  anv  point,  the  mass  of  a.  small  volnnu>  of  the  l)o(ly  snrrotmd- 
ing  the  point  (4',,  rj;,  (;)  will  Ik;  a])proxiniately  J>(t;.  rj;.  ^,)Ar,,  and  will 
surely  lie  Ijetween  tlie  limits  .V^ATj  and  7«,Ar,-,  whci'e  .!/•  and  ///■  are 
the  maximum  and  minimum  values  of  the  density  in  llie  element  of 
volunu!  A  I';.     The  total  mass  of  the  body  vvoidd  be  taken  as 

lim    y  />('t^,  ,?,,  ^,.) Ar,  =  fl>(.r,  !,,  x)>lV,  (5) 

where  the  sum  is  extended  over  the,  whole  l)0(ly.  That  the  limit  of  the 
sum  exists  and  is  in(l('])endent  of  the  uudhod  of  choice  of  tlu;  points 
(t,.  rj,.  ii)  and  of  tlie  method  of  division  of  the  total  volume  into  elements 
AT,-,  pi'ovided  ])(r,  //,  r:)  is  continuous  tiiid  the  elenuuits  Al'j  a})[)roaeli 
zero  in  such  a.  manner  that  they  be(;ome  small  in  every  direction,  is 
tolerably  tipparent. 


ON  MULTIPLE  I>^TEGRALS 


327 


The  evaluation  of  the  triple  integral  by  repeated  or  iterated  integra- 
tion is  the  inniiediate  generalization  of  the  method  used  for  the  doul)le 
integral.  If  the  region  over  which  the  integration  takes  place  is  a  rec- 
tangular parallelepiped  with  its  edges  parallel  to  the  axes,  the  integral  is 

fD(x,  ij,  z)dV  =  f  '  f''  C  'd(x,  y,  z)dxd>/dz.  (5') 

The  integration  with  respect  to  x  adds  up  the  mass  of  the  elements  in 
the  column  upon  the  hase  dydz,  the  integration  with  res})ect  to  //  then 
adds  these  columns  together  into  a  lamina  of  thickness  d::,  and  the 
integration  with  respect  to  z  finally  adds 
together  the  laminas  and  obtains  the  mass 
in  the  entire  parallelepiped.  This  could 
V)e  done  in  other  orders  ;  in  fact  the  inte- 
gration might  be  performed  first  with  re- 
gard to  any  of  the  three  variables,  second 
with  either  of  the  others,  and  finally  with 
the  last.  There  are,  therefore,  six  ecpuva- 
lent  methods  of  integration. 

If  the  region  over  which  the  integration 
is  desired  is  not  a  rectangular  parallele- 
piped, the  only  modification  which  must  be  introduced  is  to  adjust  the 
limits  in  the  successive  integrations  so  as  to  cover  the  entire  region. 
Thus  if  the  first  integration  is  with  res})cct  to  x  and  the  region  is 
bounded  by  a  surface  ./•  =  t//„(//,  r:)  on  the  side  nearer  the  //."-plane  and 
by  a  surface  x  =  i/'jO/,  -)  <>'i  tlic  remoter  side,  the  integration 

■■ ./,,  (//,  -) 

/)  (./•,  //,  ,-)  dxdijdz  =  n  {ij,  z)  dydz 

will  add  up  the  mass  in  elements  of  the  column  which  has  the  cross 
section  dydz  and  is  intercepted  between  the  two  surfaces.  The  problem 
of  adding  up  the  columns  is  ineiely  oiu^  in  dcjuble  integration  over  the 
region  of  the  //."-plane  npon  wliieh  tliey  stand;  this  region  is  the  pro- 
j^'ction  of  the  given  \-olume  upon  the  v/.v-plane.  The  value  of  tlie 
integral  is  then 


£ 


DdV=j        j  Qdydz=\        j  I  Ddxdydz. 


(5") 


Here  again  the  integiTitions  may  be  ]ierformed  in  any  ordei-.  pi'ovided 
the  limits  of  tln^  integrals  are  carefully  adjusted  to  (correspond  to  that 
(ji-der.    The  method  may  best  be  learned  by  example. 


328 


IXTEGRAL  CALCULUS 


Find  the  mass,  center  of  i:mvity.  iind  ludniei.t  of  inertia  about  the  axes  of  the 
vohnue  of  the  cylinder  x-  +  y-  —  2r//;  =  0  which  lies  in  the  hrst  octant  and  under 
paraboloid  x-  -{■  y-  ~  az,  if  the  density  be  assumed  constant.  The  integrals  to  eval- 
uate are  : 

/  J^'lin  /  i/ihn  jzdm 

m  =  f])dV.         2  =  -^ -,         y=    ,         z  =  ~ ,  •  (6) 

J  lie  in  in  ^  ' 

Ix  =  jl)  {U-  +  2-)  d  T',         ly  =  dJ  (./•■-  +  z")  dV,         7,  =  DJ^x-  +  y"-)  d  V. 

The  consideration  of  Imw  the  fii^'ure  looks  shows  that  the  limits  for  z  are  z  =  0  and 
z  —  {x-  +  y-)/a  if  the  lirst  integration  be  witii  respect  toz  ;  then  the  double  integral 
in  X  and  y  has  to  be  evaluated  over  a  semi- 
circle, and  the  lirst  integi'atio;i  is  nuire  simple 
if  made  with  respect  to  y  with  limits  y  =  0 
and  ?/  =  V2r<x  — X-.  and  linal  limits  x  =  0 
and  X  =  2n  for/.  If  the  attempt  were  made 
to  integrate  lirst  witli  respect  to  y,  there 
would  be  dilRculty  because  a  line  parallel  to 
tlie  y-axis  will  give  different  limits  according 
as  it  cuts  both  the  paraboloid  and  cylinder  i.r 
tlie  x^-plane  and  cylimler  ;  the  total  integral 
would  be  the  sum  of  two  integrals.  There 
would  be  a  similar  difficulty  with  respec: 
to  an  initial  integration  by  x.  The  order  of 
integrati(jn  should  therefore  l;a  z,  y,  x. 


.T  =  2« 


/  f  dzdydx  =  I)(         / 

=    ^    (  '    \  ,/•-  V2  I IX  -  X-  +     (2  ax  —  X^)2    ilr 

<i  J'-,     1  :i  J 

=  Ihr  f"\  (1-  cos(9)-sin-(9  +  ^^  sin*  ^'M^  -^  "  TTirD 

=  /  xdzdydx  =D  / 


II 


di/dx 


r.'.-  =  »/(!-  cos/9) 
\  dx  =  (/  sin  OiW 


./•■'  +  .'■//- 


lydx 


\  2  tix  —  X-  +  -  X  (2  ((.;:  —  ./•-)-    '/./•  =  tth^D. 

Hence  .7  =  4  n/Z.   Tlie  computation  of  the  other  integrals  may  l)e  left  as  an  exercise. 

134.  Soiiiftiines  tli-t',  rcgidii  over  wliicli  a  inulviplti  iiiti-grul  is  to  1h' 
evaluated  is  stick  that  the  evahiatioii  is  rehitivcly  siinjilc  in  one  kind 
of  coordinates  hut  eiitirtdv  impracticabk'  in  anotker  kind.  In  a(klili()n 
to  tke  rectangular  (■(xirdinates  tke  most  useful  systems  are  imlar  ru'nv- 
dinates  in  tke  plane  (for  doulile  integrals)  and  jiDlar  aiul  cylindrical 
eoiii'dinatcs  in  space  1  ini-  ti-iplc  integi-als  1.  It  lias  keen  seen  i  ;;  40)  tkat 
llic  element  of   area  or  of  \oluinf  in  tkese  cases  is 


dA 


i/nI<P, 


<l\ 


^m6iJi-<ld'l^. 


ilV  =  r.lr^l^ih:. 


ox  .MULTIPLE   INTEGRALS 


329 


(v^«,o) 


except  for  infinitesiinals  of  higher  order.  These  quantities  may  be 
substituted  in  the  douljle  or  triple  integral  and  the  evaluation  may  l)e 
made  by  successive  integration.  The  proof  that  the  sul)stitution  can 
be  made  is  entirely  similar  to  that  given  in  §§  34-35.  The  proof  that 
the  integral  may  still  Ije  evaluated  by  successive  integration,  with  a 
proper  choice  of  the  limits  so  as  to  cover  the  region,  is  contained  in 
the  statement  that  tlie  formal  work  of  evaluating  a  multiple  integi-al 
by  rei)eated  integration  is  independent  of  what  the  coordinates  actually 
represent,  for  the  reason  that  they  could  be  interpreted  if  desired  as 
representing  rectangular  coordinates. 

Find  the  area  of  the  part  of  one  loop  of  the  lemniscate  r-  —  2a-  coii2<p  wliicli  is 
exterior  to  the  circle  r  =  «  ;  also  the  center  of  p-avity  and  the  moment  of  inertia  rela- 
tive to  the  origin  under  the  assumption  of  constant  density.   Here  the  integrals  are 

A=CdA,        A2=  CxdA,        Ai;'=CydA,         I  =  I)Cr-aA,         m  =  nA. 

The  integrations  may  be  performed  first  wiili  respect  («.  Jc't)^ 

to  r  so  as  to  add  up  the  elements  in  the  little  radial 
sectors,  and  then  with  regard  to  4>  so  as  to  add  the 
sectnrs  ;  or  first  with  regard  to  <p  so  as  to  combine  the 
elements  of  the  little  circidar  strips,  and  then  with  re- 
gard U)  r  s(j  as  to  add  up  the  strips.    Thus 

IT  /  ~.,  TT 

X/> « V :!  COS- ,/,  ^-  l\       —         ^ 

"      (  rdrd4>  =   /  "  (2  a-  C(js  2,p-  a^) d<p  =  /  -  ^  ;]  -     \a-  .    .:J4:;  a-, 

TT  I  '7~  TT 

P   ,  /I  (( V  -  COS- A  2     r  .•  r- 

Ax  =2      "      /  r  cos^  .  rdnl4>  =  -  /   \2  x2 , 

_  2    „  /> ,'; [2  \'^  (1  —  2  sin-^i) '^  (7  sin  0  —  cos  0'i0]  ='^-(r  =  ..30.S  a" 

Hence  x  =  ■jTrii/i\2\''>]  —  4  tt)  =  l.lor(.  'J'he  sym- 
metry of  the  iigure  shows  that  7/  =  0.  The  calcula- 
tion of  I  may  be  left  as  an  exercise. 

(jiven  a  sphere  of  which  the  density  varies  as  the 
distaiieL'  fri)m  sumo  point  of  the  surface  :  i'ei|uired  the 
ma.-s  and  the  center  <if  gravity.  If  polar  coordinates 
^vith  tiie  origin  at  the  given  point  and  the  polar  axis 
aloULC  the  diameter  thnnigli  that  ]i(jiiil  be  assumed, 
the  tMiuatiiin  of  the  sphere  re(luces  to  ;•  =  2  a  cos  ^ 
where  a  is  the  radius.  The  center  of  gi-avity  from 
reasons  of  synnnetry  will  fall  on  the  diameter.  To 
cover  the  volume  of  the  spln.-re  /•  must  vary  from  /•  =  0 
at  the  orii:in  to  r  =  2am<  9  upon  the  sphere.  The 
jiolar  angle  must  range  from  (9  =  0  to  ^  =  I  tt.  and  the 
longitudinal  anule  froiJi  cp  ~  0  to  y  =  2  tt.    'I'hen  ^<^  =  u 


(r )  cos  00 


330  IXTECxRAL  CALCULUS 

J-.27r       /^^        ^lii  cos  e 

Jr^in       p  -         /^  ;•=■_•  a  cos  ^ 
/    "      I  L-r  ■  r  i-ofiO  ■  7--mi0drdOd4>, 

jn  =    f " "    r  ^  4  Av(*  cos-i  ^  si n  0dddcf>  =  f ' "  ^  /c'(*c/0  =  ^^^ : 


'<^=o 


4  ku* 
82  A7f= 


/-■■^'^     />.,     82A7f^        „„    .     „,„,  r ^'^32 Ay/5  ,         Giirka^ 

I  / coa'^  e  i^\\\  6ded<j)  =:   I  J0  =  — -.— • 

The  center  of  gravity  is  tlierefore  2  =  8  a/7. 

Sometimes  it  is  iiecessaiy  to  make  a  change  of  variable 
x  =  4>(h,  r),  !/  =  ^(jf,  f) 

or  X  =  4>("i  '"j  ''■)'  tf  —  ^(j'f  ''i  "■)'  ■'  ~  *^("'  ''i  "')  (^) 

ill  a  double  or  a  triple  integral.    The  element  of  area  or  of  volume  has 
been  seen  to  be  (§  63,  and  Ex.  7,  p.  135) 


clA  =  J 


■>  !/ 


dinh'     or     <IV 


] lence 


and 


C  I)  (,>;>/)</.[=   C 


I)  (j;  >/)</.[  =   I  J)(cf>,  f )  ./ 


/ 


D(,;  I/,  ^)dV=   I    D(cj>,  xp,  co) 


/• 


,/ 


■''. 

.'/, 

""l 

", 

'', 

n-) 

':,-, 

?/^ 

idu 

>', 

VJ 

.r, 

Ih 

'"^ 

", 

'", 

,■) 

dudi'dw. 


dudvdw. 


(8') 
(8") 


It  should  l)e  noted  that  the  Jacobian  ma}'  be  either  positive  or  negative 
l)ut  should  not  vanish;  the  difference  between  the  case  of  positive  and 
tlie  case  of  negative  values  is  of  tlie  same  nature  as  the  difference 
between  an  area  or  volunu^  and  tlu^  reflection  of  the  area  or  volume. 
As  the  elements  of  area  or  volume  are  considered  as  positive  Avhen 
the  increments  of  the  variables  are  positive,  the  absolute  value  of  the 
Jacobian  is  taken. 

EXERCISES 

1.  Slinw  tluvt  ((■))  are  tlie  forinuliis  for  the  center  of  gravity  of  a  solid  body. 

2.  Show  that  /,,  ~  C (if-  +  z-)diii,  J,,  =  f  (x-  +  z-)dvi,  7-  =  f  {x-  +  !/-)dm  are  the 
formulas  foi-  the  iiiomeiit  of  inertia  of  a  solid  ahoiU  the  axes. 

3.  Prove  that  the  difference  between  the  moments  of  inertia  of  a  .solid  about 
any  line  and  about  a  parallel  line  through  tlie  center  of  gravity  is  the  product  of  the 
mass  of  the  body  by  the  s(iuare  of  the  perpendicular  distance  between  the  lines. 

4.  Find  the  moment  of  inertia  of  a  body  about  a  line  thrcjugh  the  origin  in  the 
direction  determined  by  the  cMsiues  /,  7n.  n.  and  sliow  that  if  a  distance  O/-"  be  laid 
off  along  this  line  iiixci'sely  iii'iiportional  to  the  sijuare  root  of  the  moment  (jf 
inertia,  the  Im/us  of  /*  is  a:i  i  l!i    snid  with  O  as  center. 


=  1. 


ON  MULTIPLE   INTEGRALS  331 

5.  Find  tlie  moments  of  inertia  of  these  solids  of  uniform  density: 
(a)  rectangular  parallelepiped  abr,  about  the  edge  «, 

(/3)  ellipsoid  x^/a^  +  y^/b'-^  +  z^/c/^  =  1,  about  the  z-axis, 

(7)  circular  cylinder,  about  a  perpendicular  bisector  of  its  axis, 

(5)  Avedge  cut  from  the  cylinder  x^  +  y^  =  r-  hjz=±  mx,  about  its  edge. 

6.  Find  the  volume  of  the  solids  of  Ex.  5  ((8),  (5),  and  of  the  : 

(a)  tri rectangular  tetrahedron  between  xyz  =  0  and  x/a  +  y/b  +  z/c  =  1, 

((3)  solid  bounded  by  the  surfaces  y^  +  z-  =  4  (i.r,  y-  =  ax,  x  =  3  a, 

(7)  solid  common  to  the  two  equal  perpendicular  cylinders  x-  +  y~  =  «-,  x-  +  z-  =  11- 

(,)  octa...  o,  (£)•.  (;/)■>  0)'=  ,        „)  „„.,.  „r  (9%  (0-.  (I* 

7.  Find  the  center  of  gravity  in  P>x.  5  (5),  Ex.  (5  (a),  (/3),  (5),  (e),  density  uniform, 

8.  Find  the  area  in  these  cases  :         {a)  between  r  =  a  sin  2  0  and  r  =  I  n. 

(/3)  between  r-  =  2  d"  cos  2  (p  and  r  =  3^  a,       (7)   between  r  =  a  sin  0  and  r  =  6  cos  0, 
(5)  r-  =  2  a-  cos  2  0,  r  cos  0  =  1  Vs  a,  (e)  r  =  a  (1  +  cos  0),  r  =  a. 

9.  Find  the  moments  of  inertia  about  the  pole  for  the  cases  in  PvX.  8,  density 
uniform. 

10.  Assuming  uniform  density,  find  the  center  of  gravity  of  the  area  of  one  loop  : 

(a)  r- =  2  a2  cos  2  0,         (/3)  r  =  f(  (1  —  cos  0).         (7)  ?•  =  «  sin  2  0, 
(5)  r  =  asin-^l  0  (small  loop),         (e)  circular  sector  of  angle  2  ex. 

11.  Find  the  moments  of  inertia  of  the  areas  in  Ex.  10  (a),  (/3),  (7)  about  the 
initial  line. 

12.  If  the  density  of  a  spliere  decreases  unifornd\'  from  7^  at  the  center  to  I)^ 
at  the  surface,  find  the  mass  and  the  moment  of  inertia  about  a  diameter. 

13.  Find  the  total  volume  of  : 

{a)  (.r-  +  y-  +  z~)-  =  axyz,         {,3)  {x-  +  ?/-  +  z~f  =  21  a^xyz. 

14.  A  spherical  sector  is  bounded  by  a  cone  of  revolution;  find  the  center  of 
gravity  and  the  moment  of  inertia  about  tlie  axis  of  revolution  if  the  density 
varies  as  the  ?ith  power  of  the  distance  from  the  center. 

15.  If  a  cylinder  of  liquid  rotates  about  the  axis,  the  shape  of  the  surface  is  a 
paraboloid  of  revolution.    Find  tlie  kinetic  energy. 

16.  Compute  ./  i'^-'-'^,  j(''^^^^^],  J  (''^-\  and  hence  verify  (7). 

^  Vr,  0/        \/-,  0,  z)        Vr,  0,  ^/  '  ^  ' 

17.  Sketch  the  region  of  integration  and  the  curves  u  —  const.,  v  =  const.  ; 
hence  show : 

{a)     I      I         /(x,  y)dxdy  =  (      (       f{u  —  uv,  uv)  ududv,  ii  u  =  y  +  x,  y  =  uv, 

Jn    J !/_=<)  Jo    J  11  =  0 

(^)     f"  f    f{x,y)dxdy 

=  I      /  / ,      I dvdu  if  y  =  xu,  x  = , 

Jo     Jv^O  \l  +   II       1  +  M/   (1  +   »)-  l  +  « 

I' 

Jrt  (t    p  \                    y                                  n-la    p 1               ): 
/' dudv  —  I        I  "      f dudv. 
0  J,.=o   (1  +  uy~          J„    J»=i  '  (1  +  11)'^ 


3-32  INTEGRAL   CALCULUS 

18.  Find  the  volume  of  the  cylinder  r  =  2  a  cos  ^  between  the  cone  z  ~  r  and 
the  plane  z  =  0. 

19.  Same  as  Ex.18  for  cylinder  r~  =  2  ffi  cofi  2  (p  :  and  lind  the  moment  of 
inertia  about  ?•  =  0  if  the  density  varies  as  the  distance  fnini  r  =  0. 

20.  Assuming  the  law  of  the  inverse  square  of  the  distance,  show  that  the 
attraction  of  a  homogeneous  .sphere  at  a  point  outside  the  sjihere  is  as  though  all 
the  mass  were  concentrated  at  the  center. 

21.  Find  the  attraction  of  a  right  circular  cone  for  a  particle  at  the  vertex. 

22.  Find  the  attraction  of  (a)  a  solid  cylinder,  ((3)  a  cylindrical  shell  uj^on  a 
point  on  its  axis  ;  assume  homogeneity. 

23.  Find  the  potentials,  along  the  axes  only,  in  Ex.  22.  The  potential  may  be 
defined  as  ^t—'^dm  or  as  the  inteirral  of  the  force. 


24.  Obtain  the  fonnulas  for  the  center  of  gravity  of  a  sectorial  area  as 
■'^1  1   „  ,  _       ]     r*^!  1 


_         1      /--it  1  _         ]      /-•^i  1 

x  =  —   I       ~i-^coH<pd(p,         y^    -   I       -  y'^  Sill  (pd(p. 


and  explain  how  they  could  be  derived  from  the  fact  tliat  tlie  center  of  gravity  of 
a  uniform  triangle  is  at  the  intersection  of  the  medians. 

25.  Find  the  total  illumination  upon  a  circle  of  radius  a.  r)wing  to  a  light  at  a 
distance  h  above  the  center.  The  illumination  varies  inversely  as  the  square  of  the 
distance  and  directly  as  the  cosine  of  the  angle  between  the  ray  and  the  normal 
to  the  surface. 

26.  "Write  the  limits  for  the  examples  worked  in  §§  1.33  and  134  when  the  inte- 
grations are  performed  in  various  other  ni'ilcrs. 

27.  A  theorem  of  Pappus.  If  a  closed  jilane  curve  be  revolved  about  an  axis 
which  does  not  cut  it.  the  volume  generated  is  etjual  to  the  product  of  tlie  area  of 
the  curve  by  the  distance  traversal  by  the  center  of  gravity  of  the  area.  Prrive 
either  analytically  (ir  l)y  inlinitesiinal  aualvsis.  .\pjih'  to  various  figures  in  which 
two  of  the  three  quantities,  volnnie.  ;',rea.  position  of  center  of  gravity,  are  known, 
to  find  the  third.    Compare  Ex.  3.  p.  34). 

135.  Average  values  and  higher  integrals.  The  value  of  some  s])eeial 
intei'})retatioii  of  iiitcyftils  ami  otiicr  mathematical  entities  lies  in  the 
concreteness  and  suyyestiveiiess  which  would  be  lacking  in  a  ])urcly 

analytical  handling  of  the  sul)ject.     For  the  simple  integral  I  fi. '■)'/.'■ 

the  curve  ?/  =f(.r}  was  ])lotted  and  the  integrtd  was  inter])ret('d  as 
an  area;  it  w()ulil  lia\'e  liocn  i)Ossil)h'  to  remain  in  one  dimensidii  Ity 
interpreting /(,'•)  ;is  the  density  of  a  rod  and  the  integral  as  the  mass. 

In  the  case  of  the  doulile  integral  j  /(■'',  ,'/)'^A  the  eoneeption  of  den- 
sity and  mass  of  a  lamina  was  made  fundamental:  as  was  jiointed  out, 
it  is  possible  to  go  into  tlirt  (■  dimensions  tind  plot  tln'  surface  .*.'  =/'(■'',. ;/) 


ox   ]\[ULTirLE   IXTEGKALS  :]:]:] 

and  intcrpivt  the  iiitegi-al  as  a  volume.    In  the  treatment  of  the  trijtle 

integral  I  /(■'',  ;/,  z)dV  the  density  and  mass  of  a  l)ody  in  s^taee  were 

made  fundamental;  here  it  Avould  not  l)e  possible  to  })lot  u  =f(.r,  >/,  ,-j) 
as  there  are  oidy  three  dimensions  available  for  })lotting. 

Another  im})ortant  interpretation  of  an  integral  is  found  in  the  con- 
ee})ti(jn  of  (ivrroge  culiii'.  If  y^,  y.„  •  ■  •,  y,,  are  n  numbers,  the  average  of 
the  numbers  is  the  quotient  of  their  sum  by  n. 

n  n 

If  a  set  of  numl)ers  is  formed  of  ir^  numbers  y^,  and  ir„  nundicis 
y.„  ■••,  and  v,^  numbers  y„,  so  that  the  total  number  of  the  luuubers 
is  v\  +  yr,  +  •  •  •  +  ("■„,  the  average  is 

The  coefficients  >r^^  v/-,.  •  • -, /r^ .  or  any  set  or  numbers  which  are  ])i'o- 
])ortional  to  them,  are  called  the  ir('ifilif.'<  of  y^.  y.,,  •••,  y„.  These  defi- 
nitions of  average  will  not  a])])ly  to  finding  the  average  of  an  inhnite 
number  of  nund)ers  because  the  dciiondnator  n  would  not  be  an  arith- 
metical ninuber.  Hence  it  would  not  l)e  possibh»  to  apply  the  definition 
to  fiiuling  the  average  of  a  function  f(y)  in  an  interval  .>Vj  ^  ./•  ^  x^. 

A  slight  change  in  the  })oint  of  view  will,  however,  lead  to  a  deh- 
nitiou  for  t]if  in-i'i-di/c  vdlin;  of  a  funcfinn.  Sup])ose  that  the  interval 
;%  =  .'■  =  .'"j  is  divided  into  a  numbei-  of  intervals  A.c,-,  and  that  it  be 
imagined  tliat  the  number  of  values  of  //  =/'(■'■)  in  the  interval  Xr. 
is  pro|)ortional  to  the  length  of  the  interval.  Then  the  quantities 
A.r;  would  be  taken  as  the  weights  of  the  values  _/'(^,)  and  the  average 
would  be  „.,.^ 

J     or  better      y  =  — ---,. (10) 

/      dx 


2A,/\. 


by  ])assing  to  the  limit  as  the  A,>-,'s  a}»proach  zero.    Then 

rf(:r)dx 

II  =  '^-^^^—^ or        r  'fix)  dx  =  (x^  -  x^)  7/.  (10') 

■^1       ''o  J,,, 

In    like    manner    if    z  =f(x,    //)    l)e   a    function    of    two    varial)les    oi' 
II  =,/'(.'•;  //,  z)  a  function  of  three  variables,  the  averages  over  an  area 


334  IN  TEG  HAL   CALCULUS 

or  volume  would  be  defined  bj  the  integrals 

f /(■';//)  d  A  ff(:r,!/,z)dV 

z  =  ^ and     ¥1  =  "^ — -•  (10") 

jdA  =  A  id  V  =  V 

It  should  be  particularly  noticed  that  the  value  of  the  average  is  de- 
fined v'lth  reference  to  tJie  variahlea  of  irhlcli  the  function  averaged  is  a 
function  ;  a  clicnige  of  variable  ivlll  in  general  bring  about  a  change  In 
the  value  of  the  ace  rage.    For 

if  y  =  /(,r),  J{7)  =  — ^    f    /(.-)  dx ; 

■'l  ''0J.r„ 

but  if  g=f(cf.(t)),        ^  =  -J—    f\f(<f,(t))dt; 

and  there  is  no  reason  for  assuming  that  these  very  different  expres- 
sions have  the  same  numerical  value.    Thus  let 

7/  =  ,r-,  0  ^  ,,■  ^  1^         -x  =  sin  t,         O^t^h  TT, 

The  average  values  of  x  and  y  over  a  plane  area  are 
.r  =  -    /  xdA,         /7  =  7    /  l/'^-^y 

when  the  weights  are  taken  ])roportional  to  the  elements  of  area;  but 
if  the  area  be  occupied  by  a  lamina  and  the  Aveights  be  assigned  as 
proportional  to  the  elements  of  mass,  then 


—    /  xd/ir,  y  =  —    /  , 


l/dm, 

and  the  average  values  of  .'•  and  //  are  the  coordinates  of  the  center  of 
gravity.  Thes(>  two  averages  cannot  l)e  expected  to  be  e(pial  unless  the 
density  is  (-onstant.  The  first  would  be  called  an  area-average  of  x  and 
y;  the  second,  a  mass-average  of  x  and  y.  Thi'  mass  average  of  the 
s(]uai'e  of  the  distance  from  a:  point  to  the  different  points  of  a  liimina 
would  be  -|      ^ 

r^  =  Jr  =  ~j  Ah,  =  I/M,  (11) 

and  is  defined  as  the  radius  of  gyration  of  the  lamina  al)Out  that  point; 
it  is  the  quotient  of  the  moment  of  inertia  by  the  mass. 


ox  MULTIPLE  INTEGRALS  335 

As  a  problem  in  averagt-.s  consider  the  deteniiination  of  tlie  average.value  of  a 
proper  fraction  ;  also  the  average  value  of  a  proper  fraction  subject  to  the  condi- 
tion that  it  be  one  of  two  proper  fractions  of  which  the  sum  shall  be  less  than  or 
equal  to  1.    Let  x  be  the  proper  fraction.    Then  in  the  first  case 

X  =  -   (    xdx  =  -  • 
1  Jo  2 

In  the  second  case  let  ?/  be  the  other  fraction  so  that  x  +  ?/  ^  1.  Now  if  (x,  y)  be 
taken  as  coordinates  in  a  plane,  the  range  is  over  a  triangle,  the  number  of  points 
(x,  y)  in  the  element  dxdy  would  naturally  be  taken  as  proportional  to  the  area  of 
the  element,  and  the  average  of  x  over  the  region  would  be 

jxdA        f'j''~"xdjrdy        j'\l-2y  +  y'-)dy       ^ 


dA  /      /         djyly  -2       {l-y)dy 

J  0    J  0  Jo 


3 


Now  if  X  were  one  of  four  proper  fractions  whose  sum  was  not  greater  than  1,  the 
problem  would  be  to  average  x  over  all  sets  of  values  (x,  y,  z,  u)  subject  to  the 
relation  x  +  v/  +  z  +  m  s  i.  From  the  analog}'  with  the  above  problems,  the  result 
would  be 


,.      SxAxAvA 
Inn 

IiA./'A//AzA« 


xdxdydzdu 


I  I  I  dxdydzdu 

.  =  0  J.:  =  0    J  '/  =  0  J.'-  =  0 

The  evaluation  of  the  quadruple  integral  gives  x  =  1/5. 

136.  The  foregoing  probleiu  and  otlier  problems  wliieh  may  arise 
lead  to  the  consideration  of  integrals  of  greater  inulti})licity  than  three. 
It  will  l)e  siifheient  to  nunition  tlie  case  of  a  quadrtiple  integral.  In  the 
first  place  let  the  four  variables  Ije 

X^^X^.r^,  >,^^;^^;/^,  -^^^X^X^  ,f^^u^U^,       (12) 

included  in  intervals  with  constant  limits.  This  is  analogous  to  the 
case  of  a  rectangle  or  rectangular  ])arallelepiped  for  double  or  triple 
integrals.  The  range  of  values  of  .'■,  //,  s:,  a  in  (12)  may  l)e  spoken  of 
as  a  rectangular  volume  in  four  dimensions,  if  it  be  desired  to  use  geo- 
metrical as  well  as  analytical  analogy.  Then  the  product  Aj;jAy,A,~,A//; 
would  be  an  element  of  the  region.    If 

.r,.  ^  4  ^  •'",•  +  A.'-,-,  ■  •  •,  ?/,  ^  Bj  ^  ,,.  +  A//,-, 

the  point  (^,,  r;,,  ^,-,  ^,)  would  be  said  to  lie  in  the  element  of  the  region. 
The  formation  of  a  quadruple  sum 


X- 


could  be  carried  out  in  a  nutnner  similar  to  that  of  double  and  tri})le 
sums,   and   the   sum   could   readily   lie   shown   to   have   a   linut   when 


336  INTEGIIAL  CALCULUS 

A./',-,  Ay,-, -A.-,-,  \Uf  approach  zero,  i)rovi(l(Hl_f  is  fontinuous.  The  limit  of 
this  sum  eoukl  l^e  evahiated  l)_v  itei'atcd  integration 

lim2^/;A.r,A//,A.t,A^/;  =  I     '  j     '  j    'fi.r,>j,z,v)ilu<h<J>j<lr 

^■'o     «^"o     ^-0     ^"o 

where  the  order  of  the  integrations  is  immateriaL 

It  is  possible  to  define  regions  other  than  ])y  means  of  inecjualities 
such  as  arose  above.    Consider 

I^Q'',  y,  '^,  ^0  =  0     and     F{x,  y,  z,  n)  s  0, 

where  it  may  be  assumed  that  wlien  three  of  the  four  varial)les  are 
given  the  solution  of  /'  =  0  gives  not  more  than  t\v(j  values  for  the 
foui'th.  The  values  of  ,r,  //,  re,  ti  Avliich  make  /•'  <  0  are  separated  from 
those  which  make  7'"  >  0  l)y  the  vahics  which  make  F  =  0.  If  the  sign 
of  /•"■  is  so  chosen  that  large  values  of  ./•.  //,  ,t,  n  make  /•'  })Ositive,  the 
values  whicii  give  /•'  >  0  Avill  V)e  said  to  be  outside  the  region  and  those 
which  give  F  <  0  Avill  be  said  to  l)e  inside  the  region.  The  value  of  the 
integral  of /'(■'•,  y.  '-,  "j  over  the  region  FS  0  could  be  fouiul  as 

I  j  j  /(■';  y,  '-,  «)  dudzdydx, 

where  n  —  Wj(,'-,  y,  z)  and  //  =  wjr,  //,  z)  are  the  two  solutions  of  F  =  0 
for  If  in  terms  of  .>■,  ;//,  ':,  and  wliere  the  tri])le  integral  remaining  after 
the  first  integration  must  l)e  evaluated  over  the  range  of  all  ])()ssil)lc 
values  for  (,/•,  _y,  r:).  ]\y  first  solving  for  one  of  the  othci'  variables,  the 
integrations  could  be  arranged  in  another  order  with  pro})crly  changed 
limits. 

If  a  change  of  variable  is  effected  such  as 
:r  =  (p(x',y',z',u').    y  =  \l/{x\y',z',u'),    z  =  x(''''-y%z\v/),    u  =  io(.r\  i/\  z'.  u')    (13) 
tlic  iiitei^rals  in  the  new  and  old  varial)les  are  related  by 

fffffi., y, z,  u) asayazan  =fffffi'P.  f, X, c) |./ (>:;|;|>v) i'^'^>'//'''^-""'-  Ob 

'I'he  residt  may  be  aceepteil  as  a  fact  in  \ie\v  of  its  analnuv  with  the  resuli>  (S)  jdr 
the  .simpler  cases.  A  ])roof,  however,  may  be  ,i;iven  which  will  serx'c  eijuaily  well 
as  another  way  of  establishiim-  those  results.  —  a  way  wliich  docs  not  depend  on  the 
soniewhat  loose  treatment  of  inlinitesimals  and  may  therefore  be  enn.-idered  us 
more  .satisfactory.  In  the  tirst  i>lace  note  that  frimi  the  relatimi  (•>])  nf  p.  ]:]i 
involving  Jacobians.  and  fmm  its  ^encralizatinn  to  sevei'al  \ariables.  it  appeai-s 
that  if  the  cliange  (14)  is  pnssihle  fur  each  of  two  transformations,  it  is  possible 
for  the  succession  of  the  two.     N(jw  for  the  simple  transformation 

X,  —  X.' ,         y  =  y'.         z  —  z\         a  =  "(.f'.  ,'/'.  -'.  u')  —  oj{f,  y.  z.  u').       (13') 


ox   .AIULTIPLE   IXTEGRALS  337 

which  involves  only  one  variable,  J  ~  (:w/cu\  and  here 

//(x,  ?/,  z,  v)du  =  ff{.r,  y,  z,  «')-,!'-  f^"'  =  ff{-^\  V^  ~\  «')    J   du', 

and  each  side  may  be  integrated  with  respect  to  x,  y,  z.  Hence  (14)  is  true  in  this 
case.    For  the  transformation 

X  =  ^  (x',  y',  2',  u'),  y  ^xp  (x',  ?/',  z',  «'),  2  =  x  (^',  2/',  2',  «')>  **  =  «'»  (13") 
which  involves  onlv  three  variables,  J  ( —  '  ' ,'  ~' — -)  =  J  (   /    '  "^ ,  I  and 

ffffi-'^  2/,  ^,  u)<Udydz  =ffffi'P,  ^,  X,  M)l'^  |dx'd2/'d2', 

and  each  side  may  be  inteL;rated  with  respect  to  ;/.  The  rule  therefore  holds  in 
this  case.  It  remains  therefore  merely  to  show  that  any  transformation  (13)  may 
be  resolved  into  the  succession  of  two  such  as  (13'),  (13").    Let 

•?!=•?',  yi  =  y\  Zi=z',  «|  =  a;(x',  ?/',  2',  «')=  w(X,,  ?/,,  2i,  «'). 

Solve  the  eijuation  u^  =  w(Xj,  ?/,,  2[.  »')  fi>r  a'  —  to,  (,c,,  ?/,,  2,,  ?(,)  and  write 

X  =  0(X,.  ?/,,  2j,   W,).  ?/  =  !/'  (X,.    ;/,.  2,.   CO,),  2  =  X  (■?!,  2/i,  2],   t"!),  "  =  ",• 

>v"()w  by  virtue  of  the  value  of  w^,  this  is  of  the  type  (13"),  and  the  substitution  of 
X,,  ?/j,  2i,  u^  in  it  gives  the  original  transformation. 

EXERCISES 

1.  Determine  the  average  values  of  these  functions  over  the  intervals: 

(<r)  X-,  0  s  J-  ^  10,         (/3)  sin  x.  0  ^  x  s  i  tt, 

(7)    X",  0  S  X  ^  7i,  (5)    C0S"X,  0  S  X  ^  1  TT. 

2.  Determine  the  average  values  as  indicated  : 

{(x)  ordinate  in  a  semicircle  .(-  +  y-  =;  (('-,  y  >  0,  with  x  as  variable, 

(jd)  ordinate  in  a  semicircle,  with  the  arc  as  variable, 

(7)  ordinate  in  seiniellipse  x  =  acos^,  y  =  hm\(p,  with  (p  as  variable, 

(5)  f(»cal  radius  of  ellipse,  with  equiangular  spacing  about  focus, 

( 6 )  focal  radius  of  ellipse,  with  equal  spacing  along  the  major  axis, 
(f)  chord  of  a  circle  (with  the  most  natural  assumption). 

3.  Find  the  average  liciglit  of  so  nuich  of  these  surfaces  as  lies  above  the  xy-plane  : 
(a)  X-  +  ,//-  +  2-  ^  <i-.         (A')  2  =  ('•*  -  p-x-  -  q-y'^,         (7)  e=  =  4  -  x-  -  //-. 

4.  Jf  a  m;\n"s  height  is  the  average  height  of  a  conical  tent,  on  how  nuich  of  the 
tloor  space  '■an  lie  sliiud  creel  '.' 

5.  (»btaiii  the  average  values  of  the  following: 

(it)  distance  of  a  point  in  a  square  frcjm  the  center,       (/3)  ditto  from  vertex. 
(7)  I'.istance  of  a  point  in  a  circle  from  the  center,         (5)  ditto  for  sphere, 
(c  )  ilistiuice  of  a  point  in  a  sphere  from  a  lixed  point  on  the  surface. 

6.  From  the  S.AV.  corner  of  a  township  persons  start  in  random  directions 
iK'tween  X.  and  E.  to  walk  across  the  township.  What  is  their  average  walk  ■' 
Which  has  it  ? 


338  INTEGRAL  CALCULUS 

7.  On  each  of  the  two  legs  fif  a  right  triangle  a  point  is  selected  and  the  line 
joining  them  is  drawn.  Show  that  the  average  of  the  area  of  the  square  on  this 
line  is  }  the  square  on  the  hypotenuse  of  the  triangle. 

8.  A  line  joins  two  points  on  opposite  sides  of  a  S(juare  of  side  n.  What  is  the 
ratio  of  the  average  square  on  the  line  to  the  given  s(iuare  ? 

9.  Find  the  average  value  of  the  sum  of  the  squares  of  two  proper  fractions. 
What  are  the  results  for  three  and  for  four  fractions  ? 

10.  If  the  sum  of  n  proper  fractions  cannot  exceed  1,  sliow  that  llie  average 
value  of  any  one  of  the  fractions  is  l/(n  +  1). 

11.  The  average  value  of  the  product  of  k  proper  fractions  is  2--''. 

12.  Two  points  are  selected  at  random  within  a  circle.  Find  the  ratio  of  t^c 
average  area  of  the  circle  described  on  the  line  joining  them  as  diameter  to  tlie 
area  of  the  circle. 

13.  Show  tliat  ./  =  r^  sin^  0  sin  (p  for  the  transformation 

X  =  r  cos  0.      y  =  y  sin  6  cos  <p.      z  =  r  sin  ff  sin  0  cos ;/',      u  =  r  sin  ff  sin  <p  sin  ■^, 

and  prove  tliat  all  values  of  x.  y.  z.  u  defined  by  x-  +  y-  -\-  z-  -\-  u-  ^  (i~  are  covered 
by  the  range  0  ^  r  ^  a,  0  ^  ^  ^  tt.  0  ^  0  ^  tt,  0^\p  ^'Itt.  What  range  will 
cover  all  positive  values  of  x.  y.  z.  u  ? 

14.  The  sum  of  the  squares  of  two  proper  fractions  cannot  exceed  1.  Find  the 
average  value  of  one  of  the  fractions. 

15.  The  same  as  Ex.  l-i  where  three  or  four  fractions  are  involveil. 

16.  Xote  that  the  solution  of  n^  =  oj{x^,  y^.  z^.  u')  for  u/  =  w^f.r^.  //j.  z^.  h,) 
requires  that  cw/cu'  shall  not  vanish.  Show  that  the  hypotliesis  that  J  dnes  imt  van- 
ish in  the  region,  is  sufficient  to  sIkav  that  at  and  in  the  neigliburlnKid  of  larli  point 
(.r,  y,  z,  u)  there  nuist  be  at  least  one  of  the  10  tlerivatives  of  0.  ■^.  y^.  a>  by  ./•.  y.  z.  u 
which  does  not  vanish  ;  and  thus  cijmplete  the  proof  df  the  text  that  in  case  ./  ^  0 
and  the  16  derivatives  exist  and  are  contiimous  the  change  of  variable  is  as  given. 

17.  The  intensity  of  light  varies  inversely  as  the  square  of  t!ie  iHstancc  Find 
tlie  average  intensity  of  illumination  in  a  hemispherical  dome  lighted  Ijy  a  lamp 
at  the  top. 

18.  If  the  data  be  as  in  Ex.  12.  p.  3ol,  find  the  average  density. 


17/ 


137.  Surfaces  and  surface  integrals.  Consider  a  surface  wliieh  has 
at  each  ])()iiit  a  tang-eiit  plane  tliat  changes  cdntin- 
uotisly  from  jxiint  to  point  of  the  surface.  ('onsi(h'r 
also  the  projection  of  tlie  surface  ttpon  a  ]ilane.  say 
the  .ry-plane,  and  assume  that  a  line  perpendicular 
to  the   plane   ctits   the   stirface   in   only   one   ]>oint.  /  |     ;  1' 

Over  any  element  '/.  1  of  the  ])rojection  there  will      /  ^dA 

1)6   a   small  jtortion   of  the   sui'face.    If  this   small 

portion  were  plane  and  if  its  ncnaual  made  an  an^le  y  with  the  .'.--axis, 
the  area  of  the  surface   (}).  1G7)  woidd  l)e  to   its  projection  as  1  is  to 


ox  MULTIPLE  LN^TEGRALS 


339 


cos  y  and  avouIcI  be  sec  yd  A.    The  value  of  cos  y  may  be  read  from  (9) 
on  page  96.    This  suggests  that  the  quantity 


-/--"■-ir[^-(ij-(i)i 


dxdy 


(15) 


be  taken  as  ilw  definition  of  tlte  area  of  the  surface,  Avhere  the  douljle 
integral  is  extended  over  the  projection  of  the  surface  ;  and  this  defi- 
nition will  be  ado})ted.  This  detinition  is  really  dependent  on  the 
particular  ])lane  upon  which  the  surface  is  projected ;  that  the  value  of 
the  area  of  the  surface  would  turn  out  to  be  the  same  no  matter  what 
plane  Avas  used  for  projection  is  tolerably  apparent,  but  will  be  }iroved 
later. 

Let  the  area  cut  out  of  a  heniisplicre  by  a  cylinder  upon  the  radius  of  the 
hemispliere  as  diameter  be  evahuited.    Here  (or  by  ireometry  directly) 


X-  +  y-  +  ^- 


cz 


cy 


dydx. 


J    [_        z-       z- J  J.,=f>J,,  =  i)  s  (I- —  X- —  y'^ 

This  inteirral  may  be  evaluated  directly,  but  it  is  better  to  transform  it  to  polar 
coordinates  in  the  plane.    Then 


'\^ 


in 


f't"         r  "  cos  <]>  (I  /I  *  IT 

8  =  2/  I  — =z:=  rilrd<i>  =  2  I        d-  (1  -  sin  0)  d<p  =  (tt  -  2)  «2. 

^.>  =  0  -',•=(1       \il-  —  /■-  -^tJ 

It  is  clear  that  tin-  half  area  which  lies  in  tlie  fii'st  r)ctant  could  be  projected  upon 

the  .f^-plane  and  tints  evaluated.     The  n-iziun  over  wliicli  the  integration  would 

extend   is    tliat,    between   .f-  +  z-  =  a-  and   the  ]iri)jectiiin 

Z-  +  ((/  =  (/'-'   (if    the   curve    of    interset-tion    of   the  .sjihere 

and  cylinder.    The  projection  could  also  be  made  on  the 

?/2-plane.    If  the  area  of  tiie  cylinder  lietween  z  =  0  and 

the  sphere   were   desireil,    projt'rtion    on   z  =  0   would    be 

tiseless.  ]irojection   on  x  =  0  W(juld    be   inv(jiveil  owing  to 

tlie  overlapping  of  the  projection  (jn  itself,  but  projection 

on  (/  =  0  would  lie  entireh'  feasible. 

To  sliow  that  tlie  detinition  of   area  does  not  depend,      i^ 
except  ai)parent]y.  upon  the  i)lane  of  projection  consider 

any  second  plane  whicli  makes  an  anule  0  with  tlie  first.    Let  the  line  of  intersec- 
tion l^e  the  ^-axis  ;  then  from  a  figure  the  new  coordinate  x'  is 


/  (-OS  d  +  z  sin  (9.  1/ 
ilxdy 


y- 


in<l     ./ 


(■'■'.  .'/) 


\x.  y)        IX 
(.'•.  //)  dx'dg        c c  dx'dy 


-\ sni  I 

ix 


,  ^  pr'txny  _  rr  r  (■'".  //)  ax  ay  _  ff  dxdy 

^JJ    cn^y'jJ       (,/■■.//)   cos-/  "  Jj   COS  7  (cos  6/ -hy;  sin  6*)' 

It  reinaiiis  to  show  that  the  deiKJiiiinator  cos  7  (cos  (9  +  y^  sin  6^)  -:  C0S7'.     Referred 
to  tlie  original  axes  the  direction  cosines  of  the  ntjrmal  are   —  p -,  —  ij  ;].  and  of 


340  ixtp:gil\l  calculus 

the  z'-axis  are  —  sin  ^  :0  :  co.s^.  Tlie  cosine  of  the  ani^de  between  these  lines  is 
therefore 

r»sin  ^  +  0  +  cos(9       n  sin  ^  + cos  (9  ,        r.  .     /,^ 

cos  7   = ^^ —  = =  cos  7  (cos  (9  +  j/sm  B). 

Hence  tlie  new  form  of  the  area  is  tlie  integral  of  sec  7'tZ.  1 '  ami  (Mnials  the  old  form. 

The  integrand  (lS=seGydA  is  called  f//p  cJcincnf  of  xurfacf.  There 
are  other  forms  such  as  dS  =  sec(r,7i)  r-nin  6 /9'f(f),  where  (V.  ii)  is  the 
angle  between  tht?  radius  vector  and  tlie  normal;  hut  they  ai\'  used 
comparatively  little.  The  possession  of  an  ex])ression  lor  tlie  element 
of  surface  affords  a  means  of  comptiting  (n-cnnjcs  arcr  suffi/rcs.  for  if 
11  =  it(.r,  //,  rS)  he  any  function  of  (./■,  //,  ,^'),  and  .-:;  =f{.r,  1/)  any  surface, 
the  integral 

will  be  the  average  of  u  over  the  surface  S.  Tlius  the  average  height 
of  a  hemisjihere  is  (for  the  surface  averagej 

Z  =  -r -,     \    r:<IS  =  ;    I    I    -.'  •  -  (7.rdi/  = ,  •  Trri-  =  ~  ; 

whereas  the  average  height  over  the  diametral  ])lane  woitld  lie  2/3. 
This  illustrates  again  the  fact  that  the  value  of  an  average  de})ends 
on  the  assum])tion  made  as  to  tlie  weights. 

138.  If  a  surface  ::  =/'('',  //)  l»e  divided  into  elements  A-S'-,  and  the 
function  u{x,  ij,  z)  \)v  formed  for  any  jjoiiit  (t,-.  rj;,  t,;)  of  the  element, 
and  the.  sum  2/^,A.s'-  1h'  extended  over  all  the  elements,  the  limit  of 
the  sum  as  the  elements  l)ec()me  small  in  e\'cry  direction  is  defined 
as  the  siirfiicr  Inti'gi'dJ  of  the  function  over  tlie  surface  and  may  be 
e\'aluated  as 


\\ux^a(^;,  rj;,  C,-U.SV=  f  >'(■'■    >/,  rS)'lS 


-If' 


[■'',  ,'/,-./'(■'■, .'/)]  ^  '•  -I-./'" +.r;;- '/.'•'///.     (17) 


Tliat  the  sum  a|)])roaclics  a  limit  iiHh'jicndciit  iv  of  liow  (t,.  7;,.  ^,)  is 
chosen  in  A>',- and  how  AN-  aitproachcs  zero  foUows  from  tlic  fact  tliat 
the  element  /'{i/,  rj;.  C,)-^'"'',-  "1  the  sum  diftVi's  uiiiioi-mly  fi'oiii  tlic 
integrand  of  Ihc  doulilc  integral  by  an  iiilinitcsimal  o!'  higlin-  i  i-ilci-. 
provided  //(.'•.  //,  ."-)  be  assuiiuMl  continuous  in  (./•.  //.  x)  for  points  near 
tlie  sui'face  and  Vl  +,/','+,/'„'"  ^"'  coutiijiious  in  (./■.//)  over  the  sui'facc. 
for  many  ]iur]ioses  it  is  more  conx-ciiiciir  to  take  as  the  noi-mal 
form    of    tlie    inte>j-rand    of    a    surface    iiilcL;ral,    instead    of    /"/S.   the 


ox  MULTIPLE   INTEGRALS 


341 


product  7'  cos  ydS  of  a  function  R  (x,  y,  z)  by  the  cosine  of  tlie  in- 
clination of  the  surface  to  the  .r-axis  by  the  element  dS  of  the  surface. 
Then  the  integral  may  be  eyaluated  orcr  citlmr  side 
of  the  surfact^ ;  for  7?  (,/•,  y,  ,^)  has  a  definite  value 
on  the  sui-face,  dS  is  a  positive  quantity,  but  cosy 
is  positive  or  negative  according  as  the  normal  is 
drawn  on  the  upper  or  lower  side  of  the  surface. 
The  value  of  the  intey'ral  over  the  surface  will  be 


I    li  (./•,  _?/,  ?:)  cos  yds  =   I  i  lldi'd iJ 


according  as  the  evaluation  is  made  ov(U'  tlie  upper  or  lower  side.  If 
the  function  7i  (,/•,  y,  -.)  is  c(jntiuuous  over  the  surface,  these  integrands 
will  be  finite  even  when  the  siu'face  becomes  perpendicular  to  the 
,r//-plane,  which  might  not  be  the  case  with 
an  integrand  of  the  form  u{.i',  y,  z)dS. 

\\\  integral  of  this  sort  may  be  evaluated 
over  a  closed  surface.  Let  it  be  assumed 
that  the  surface  is  cut  by  a  line  parallel  to 
the  ,r-axis  in  a  finite  numl)er  of  points,  and 
for  (convenience  let  that  numlx^r  be  two.  Let 
the  normal  to  the  surface  l)e  taken  con- 
stantly as  the  exterior  normal  (some  take 
the  interior  normal  with  a  resulting  change 
of  sign  in  some  foi'nndas),  so  that  for  the 
u])])er  jiai't  of  the  sui-face  c;)s  y  >  0  and  for 
the  lower  part  cos  y  <  0.  Let  ::  =f^(,r,  y) 
and  ,-.■  =/'(,/',  y)  l)e  the  u}»per  and  hnver  values  of  r;  on  the  surface.  Then 
the  exterior  integral  over  the  closed  surface  Avill  have  the  form 

Jn  cos  y./.S  ^-JJn  [,r,  y,f^ (,r,  //)] r/.v/.y  -JJj!  [,;  y,f^r,  y)-]d.rdy,  (18') 

where  the  double  integrals  'avc  extended  over  the  area  of  the  ])rojection 
of  the  sui'face  on  the  .'-//-plane. 

Lrom  this  form  of  the  surface  integral  over  a  closed  surface 
it  ap])ears  that  a  surface  integral  over  a  closed  surfa<-e  may  be  ex- 
])iessed  as  a  volume  integral  over  the  volume  inclosed  by  the  surface.* 

*  Certain  restrictions  upmi  tlie  fmictioTis  and  ileriviitives,  as  vetiards  tlieir  becoming 
inliiiite  anil  the  like,  must  hold  upon  and  within  the  surface.  It  ^\ill  he  cxuite  .sufficient 
if  tlie  functions  and  derivatives  renuiiu  Unite  and  continuous,  but  such  extreme  conditions 
ar(»  by  uo  means  necessary. 


2  =/l(^>  OT 

dxdy. 

z  =ff,{x,  y) 


342  INTEGRAL   CALCULUS 

For  by  the  rule  for  integration, 

/  I  /  T^  dzdxihj  =  j  hi  (■'•,  ;'/,  ^) 

Hence  /  7Z  cos  y^/N  =  I    ^- (/U 

or  j  jnd.rdi/  =  I     j    h  (ixdydz 

if  the  symbol  O  be  used  to  designate  a-  closed  surface,  and  if  the  double 
integral  on  the  left  of  (19)  be  understood  to  stand  for  either  side  of 
the  equality  (18').    In  a  similar  manner 

/7'  cos  ads  —  I  I    Pdifdz  —   \\\    w-  dxdydz  =  \     —  dV, 
C  Q  cos  lids  =  11   Qdxdz  =  CnYdydxdz  =  f  ^  ^^^^- 

Then     \  (P  cos  a  +  Q  cos  /3  +  7.*  cos  y)dS=  j  ("^  +  ;^  +  g^ )  ^^^ 

,   '"        ^.  ,  (20) 

or   rr  (7^///r/,^  +  Qdzdx  +  /i-rfoy/v/)  =  /  /  /  (^!  +  ?^^  +  ^  )  '''•^''^/M^ 

follows  immediately  by  merely  adding  the  three  equalities.  Any  one  of 
these  equalities  (19),  (20)  is  sometimes  called  Gfn/ss's  Foniniht,  some- 
times Grcciis  I^i'iiiiii((,  sometimes  the  din'riicncc  funnula  owing  to  the 
interpi'etation  l)elow. 

The  iiiter])retation  of  Gauss's  J^'oi'mula  (20)  by  vectors  is  im])ortant. 
From  the  viewpoint  of  vectors  the  element  of  surface  is  a  vec^tor  </S 
directed  along  the  exterior  normal  to  the  surface  (§  76).  Construct  the 
vector  function 

F(,r,  7/,  .^)  =  i7>(.T,  y,  r;)  +  VH-'',  Ih  ^^  +  kTZ  (.r,  y,  z). 

Let      r/S  =  (i  cos  a  +  j  cos  /3  +  k  cos  y)(IS  =  idS^.  +  jr/.S'J  +  krAs\, 

where  dS,.,  dS,^,  dS,  are  the  ])rojecti()ns  of  dS  on  the  coordinate  plancc 

Then  1'  cos  adS  +  Q  cos  fSdS  +  R  cos  ydS  =  F.(/S 

and  [ I  (Pdyd::  +  (^/./v/,-;  +  /!</xdf/)  =  (  Y.dS, 


where  <fS_^.,  dS,^,  dS^  have  l)een  replaced  \)\  the  elements  <h/dr:,  drdr:,  dxdy, 
whi(th  would  be  used  to  evaluate  tlie  integrals  in  rectangular  coordinates, 


ON  ]\IULTIPLE  INTEGEALS  843 

without  at  all  iniplying-  that  the  projections  dS^,  dS,^,  dS^  are  actually 
rectangular.    The  combination  of  partial  derivatives 

^  +  |i^  +  |^  =  divF  =  V.F,  (21) 

ex       Ci/       cz  ^     -^ 

where  V.F  is  the  symbolic  scalar  product  of  V  and  F  (Ex.  9  below),  is 
called  the  dirergeyice  of  F.    Hence  (20)  becomes 


rdivFf/r=  C\.YdV=  fF.dS. 


(20') 


Now  the  function  F(,/;,  y,  z)  is  such  that  at  each  point  (/,  y,  z)  of  space  a  vector 
is  (U'tined.  Such  a  function  is  seen  in  the  velocity  in  a  moving  fluid  such  as  air  or 
water.  The  picture  of  a  scalar  function  u  (x,  ?/,  z)  was  by  means  of  the  surfaces 
u  =  const.;  the  picture  of  a  vector  function  F(j,  y.  z)  may  be  found  in  the  system 
of  curves  tangent  to  the  vector,  the  stream  lines  in  the  fluid  ,„ 

if  F  be  the  velocity.    For  the  inunediute  purposes  it  is  better  ^^ 

to  consider  the  function  F(.r,  ?/,  2)  as  the  flux  7>v,  the  prod- 
uct of  the  density  in  the  fluid  by  the  velocity.  With  this 
interpretation  the  rate  at  which  the  fluid  fluws  through  an 
element  of  surface  dS  is  Dw-dS  =  F-JS.  F<ir  in  the  time 
lit  the  fluid  will  advance  along  a  stream  line  by  the  amount 
vdt  and  the  volume  of  the  cylindrical  volume  of  fluid  which  advances  through  the 
surface  will  be  V'dSdt.  Hence  ^iJy-dS  will  be  the  rate  of  diminution  of  the  amount 
of  fluid  within  the  closed  surface. 

As  the  amount  of  fluid  in  an  element  of  volume(/T"  is  l)dV.  the  rate  of  diminution 
(if  the  fluid  in  tlic  element  of  volume  is  —  cD/ct  where  cD/ct  is  the  rate  of  increase 
of  the  density  1>  at  a  point  witliin  the  element.  The  total  rate  of  diminution  of  the 
amount  of  fluiii  witliin  the  whole  volume  is  therefore  —  'ZcD/ctdV.  Hence,  by 
virtue  of  tlie  principle  of  the  indestructibility  of  matter, 

f  F.tZS  ^  f  Dy.dS  =  -  C'^dV.  (20") 

Now  if  i\r.  Vf,.  v-_  lie  the  components  of  v  so  that  P  =  Dvx,  Q  =  Dv,,.  E  =  Dv^  are 
the  components  of  F.  a  coiuparison  of  (21).  (20'),  (20")  shows  that  the  integrals  of 
—  cl)/ct  and  div  F  ai'e  always  equal,  and  hence  the  integrands, 

cl)      (P      cQ      ni  _clhj:      cBv,,      cJh\ 
ct        C.C       cy        cz         ex  cy  cz 

are  equal  :  that  is.  the  sum  Pjl  +  (/^  +  B'^  represents  the  rate  of  dimiiuition  of 
density  when  iP  +  jQ  +  kli  is  the  flux  vector;  this  combination  is  called  the 
divergence  of  the  vector,  no  matter  wliat  the  vector  F  really  represents. 

139.  Xot  only  may  a  surface  integral  be  stepped  up  to  a  volume 
integral,  but  a  line  integral  around  a  closed  curve  may  be  stcpjied  up 
into  a  surface  integral  over  a  surface  which  spans  the  curve.    To  begin 


344 


INTEGRAL  CALCULUS 


with  the  simple  case  of  a  line  integral  in  a  plane,  note  that  by  the 
same  reasoning:  as  above 


(22) 


This  is  sometimes  called  Green's  Lemma  for  the  plane  in  distinction 
to  the  general  Green's  Lemma  for  space.  The  oppo- 
site signs  must  be  taken  to  preserve  the  direction 
of  the  line  integral  about  the  contour.  This  result 
may  be  used  to  establish  the  rule  for  transforming  a 
double  integral  by  the  change  of  variable  x  =  ^  (u,  r), 
y  =  ^Oh  v).    For 


A' 


}  ■^  L       CU  CO 


f/y 


(.^\- 


cu  \    cr 


c  /     CiJ 

y—     X  T— 

cc\    cu 


J  J    \cu  CO       Co-  cu  I 


dude 


J \ dud  I 


(The  double  signs  have  to  l)e  introduced  at  first  to  allow  for  the  case 
where  J  is  negative.)  The  element  of  area  dA  ^=\J\dudv  is  therefore 
established. 

To  obtain  the  formula  for  the  conversion  of  a        ^ 
line    integral    in    s})ace   to  a  surface   integral,   let 
P(x,  ?/,  ,'i)  be  given  and  let  r:  =f(x,  //)  be  a  surface         oj- 
s])uniiing  the  closed  curve  O.     Then  by  virtue  of     / 
z  =/(.'■,  y),  the  function  J^(x,  //,  -■)  =  P^(x,  i/)  and 

where  O'  denotes  the  projection  of  O  on  the  .>7/-]jlane.  Xow  the  final 
diiuhle  integnd  may  be  transformed  by  the  introduction  of  the  cosines 
of  the  normal  direction  to  ;:  =f{,r^  ijy 

cos  ^ :  cos  y  =  —  Y  :  1,      dxd ,j  =  cos  ydS,    iplxdij  =  —  cos  ^dS  =  —  dxdz. 


ox  MULTIPLE  IXTEGEALS 


345 


Then 


^(i-i)--ir(i 


cP  cP 

dxdz ; —  dxd  If 


£ 


Pdx. 


If  this  result  and  those  obtained  by  permuting  the  letters  be  added, 

UPdx  +  Qdij  +  Ud-:) 
^  o 


■■II 


cR 


cQ\  ,    ,        (cP      cR 

c::  \  cz       ex 


ex        ClJ 


dxdy 


(23) 


This  is  known  as  Sfol-cs's  Fonmihi  and  is  of  especial  importance  in 
hydromechanics  and  the  theory  of  electromagnetism.  Note  that  the 
line  integral  is  carried  around  the  rim  of  the  surface  in  the  direction 
which  ap})ears  positive  to  one  standing  u})on  that  side  of  the  surface 
over  which  the  surface  integral  is  extended. 

Again  the  vector  interpretation  of  the  result  is  valuable.    Let 

F  (x,  y,  -:)  =  iP  (,/•,  j/,z)+iCl  (x,  >/,  z)  +  kR  (x,  f/,  z), 


Then 


,  .^       j'cR       cQ\        . /cP       cR\       ,    /cQ       cP\ 
curl  F  =  lU--—    +JT--  y-    +  k    v^  -  V- 
\ct/        cz  I  \c::        ex  I  \cx        cijj 

CF.dT=  Ccnv\F.dS=  f' 


VxF.r/S, 


(24) 


(23') 


where  V^F  is  the  symbolic  vector  })roduct  of  V  and  F  (Ex.  9,  l»elow), 
is  the  form  of  Stokes's  Formula;  that  is,  the  line  integral  of  a  vector 
around  a  closed  curve  is  equal  to  the  surface  integral  of  the  curl  of  the 
vector,  as  defined  l)v  (24),  around  any  surface  Avhich  spans  the  curve. 
If  the  line  integral  is  7.ero  about  every  closed  curve,  the  surface  inte- 
gral must  vanish  over  every  surface.  It  follows  that  curl  F  =  0.  For 
if  the  vector  curl  F  failed  to  vanish  at  any  i)oint,  a  small  plane  sur- 
face dS  perpendicular  to  the  vector  might  be  taken  at  that  point  and 
the  integral  over  the  surface  would  Ix'  approximately  [curl  F\(/S  and 
would  fail  to  vanish,  —  thus  contradicting  the  hypotliesis.  Xow  the 
vanishing  of  the  vector  curl  F  rcfpiires  the  vanishing 

7^;  -  q:  =  0,       y^:  -  y^;  =  o,       q;  -  p;  =  o 

of  each  of  its  comj^jnents.  Thus  may  l)e  derived  the  condition  that 
pdx  +  Qd//  +  /.'(/,"  be  an  exact  differential. 

If  F  be  interpreted  as  the  velucity  v  in  a  fluid,  tlie  inte,i.Tal 

Tv-r^r  =  f^'-'dx  +  v,jdy  -f  v^dz 

of  the  coniprinent  of  the  velocity  alonir  a  curve,  whether  fipen  or  closed,  is  called 
the  circulation  o(  the  fluid  ali.iiii,^  the  curve ;  it  might  be  more  natural  to  deflne 


346  I^'TEGRAL  CALCULUS 

the  integral  of  the  flux  Dy  along  the  curve  as  the  circulation,  but  this  is  not 
the  convention.  Now  if  the  velocity  be  that  due  to  rotation  witii  the  angular  veloc- 
ity a  about  a  line  through  the  origin,  the  circulation  in  a  clused  curve  is  readily 
computed.    For 

V  =  axt.  f  V'dT  =  faxr-f/r  =  (  a-rxdr  =  a.  |  rxiZr  =  2a.A. 

The  circulation  is  therefore  the  product  of  twice  the  angular  velocity  and  the  area 
of  the  surface  inclosed  by  the  curve.  If  the  circuit  be  taken  indefinitely  small,  the 
integral  is  2  a-dS  and  a  comparison  with  (23')  shows  that  curl  v  =  2  a;  that  is.  the 
curl  of  the  velocity  due  to  rotation  about  an  axis  is  twice  the  angular  velocity  and 
is  constant  in  magnitude  and  direction  all  over  space.  The  general  motion  of  a 
fluid  is  not  one  of  uniform  rotation  about  any  axis;  in  fact  if  a  small  element  of 
fluid  be  considered  and  an  interval  of  time  St  be  allowed  to  elapse,  the  element 
will  have  moved  into  a  new  position,  will  have  been  somewhat  deformed  owing  to 
the  motion  of  the  fluid,  and  will  have  been  somewhat  rotated.  The  vector  curl  v. 
as  deflned  in  (24),  may  be  shown  to  give  twice  the  instantaneous  angular  velocit}' 
of  the  element  at  each  point  of  space. 

EXERCISES 

1.  Find  the  areas  of  the  following  surfaces  : 

(a)  cylinder  x-  +  y-  —  ox  =  0  included  by  the  sphere  x-  +  y-  +  z-  =  ft-. 

{13)  x/n  +  y/h  +  z/f  —  1  in  first  octant.      (7)  .r-  -\-  y-  -\-  z-  =  a-  above  r  —  a  cos7i0. 

(5)  spliere  /-  -[-  y-  +  z-  =  a-  above  a  square  \x  ^  h.  \y\  ^  h.  h  <  l\  2  a. 

( e )  z  —  xy  over  x-  +  y-  =  a'-.         ( f)  2  az  —  x-  —  y-  over  r-  =  a-  cos  0, 

(7;)  z-  +  (x  cos  a  +  ysin  a)-  —  a-  in  first  octant,         {&)  z  =  xy  (jver  r-  =  cos  2  0, 

( I )  cylinder  x-  +  7/-  =  a-  intercepted  by  equal  cylinder  y-  +  z-  =  a-. 

2.  Compute  the  following  superficial  averages: 

(a)  latitude  of  places  north  of  the  ecpuitor,  Ans.    32j"j°. 

(/3)   ordinate  in  a  right  circular  cone  }i-{x-  +  ?/-)  —  fi-{z  —  h)-  =  0. 

(7)   iiiununation  of  a  hollow  spherical  surface  by  a  light  at  a  point  of  it, 

(5)  iiiununation  of  a  hemispherical  surface  by  a  distant  light. 

(e)  rectilinear  distance  of  points  north  of  equator  from  nortli  pole. 

3.  A  theorem  of  Pappus:  If  a  closed  or  open  plane  curve  be  revolved  about  an 
axis  in  its  plane,  the  iirea  of  the  surface  generated  is  eijual  to  the  product  of  the 
length  of  tiie  curve  by  the  dist:ince  described  liy  the  center  of  gravity  of  tlie  curve. 
The  curve  sluiU  not  cut  the  axis.  I'rove  eithei-  analytically  rir  by  infinitesimal 
analysis.  Apply  to  various  figures  in  wliich  two  of  the  three  (juantities.  lengtli  of 
curve,  area  of  surface,  position  of  center  of  gravity,  are  known,  to  find  the  third. 
Compare  Ex.  27,  p.  332. 

4.  The  surface  integrals  are  to  be  evaluated  over  the  closed  surfaces  by  express- 
ing them  as  volume  integrals.    Try  also  direct  caiculatiim  : 

(a)   j  j  (x-dydz  -\-  xydxdy  +  xzdxdz)  over  the  spherical  siu'face  x-  +  y-  +  z-  =  (t-. 

(J)   11  {x-dydz  +  y'-dxdz  +  z-dxdy),  cylindrical  surface  x-  +  y"  --  a-,     z  —  ±h. 


ox  MULTIPLE  INTEGRALS  347 

(y)  I  /  [i-'^"  ~  yz)dydz  —  2xydxdz  +  dxdij]  over  the  cube  0  ^  x,  ij,  z  ^  a, 
(5)   fCxdydz  =   C C ydxdz  =  C C zdxdy  =  +  C C {xdydz  +  ydxdz  +  zdxdij)  =  F, 

(e)  Calculate  the  line  inteurais  of  Ex.  8,  p.  297,  around  a  closed  path  formed  by 
two  paths  there  <riven,  liy  applying  Green's  Lemma  (22)  and  evaluating  the  result- 
ing double  integrals. 

5.  If  X  =  <P]{u,  v),  y  =  (p.,{it,  v),  z  =  (p^{u,  v)  are  the  parametric  equations  of  a 
surface,  the  direction  ratios  of  the  normal  are  (see  Ex.  15,  p.  135) 


3S/3:coS7  =  /,:/.:J3     if     /,  =  / /^i±l^^i±^'i 

\      u,v      I 


COS  ix  :  cos  I 
Show  P  that  the  area  of  a  surface  may  be  written  as 


.S-  =  ff  — '  ^  ^  '  -  "^  '  •'  dxdy  =   ff  Vji"  +  Ji  +  -T'i  dudv  =   ff  -yjEG  -  FHudv, 

where  E  =^  i^\'  ,        G'  =  V  i^A'  ,        F  =  V  ^  '-^ , 

and  dn-  =  Edu~  +  2  Fdudv  +  Gdv'-. 

vShow  2^  that  the  surface  integral  of  the  first  type  becomes  merely 

jy/(,f,  y,  z)  i^ccrh'ly  =  fff{4'x,  4>2,  'P-d  ^^^'^  -  l-'-dudv, 

and  determine  tlie  integrand  in  the  case  of  the  developable  surface  of  Ex.  17,  p.  143. 
Show  3^  that  if  x  =/,($,  rj.  f),  y  =^  f„{^,  ?;,  f),  z  =/;;(?.  v-  f)  i'"^  '^^  transformation  of 
space  which  transforms  the  above  surface  into  a  new  surface  ^  =  f^in.  v).  rj  —  i/'o(m,  v), 
i'='/'3("»  '-•),  then 

Show  4'^  that  the  .surface  integral  of  the  second  type  becomes 

where  the  integration  is  now  in  terms  of  the  new  variables  ^,  rj,  f  in  place  of  x,  y,  z. 
Show  5°  that  when  E  =  z  the  double  integral  above  may  be  transformed  by 
Green's  Lemma  in  such  a  manner  as  to  establish  the  formula  for  change  of  variables 
in  triple  integrals. 

6.  Show  that  for  vector  surface  integrals  I  UdS  —  j  Vl'dV. 

7.  Solid  angle  ns  a  mrface  integral.  The  area  cut  out  from  the  unit  si)here  by  a 
cone  with  its  vertex  at  the  center  of  the  sphere  is  called  the  solid  angle  u)  subtended 
at  the  vertex  of  the  cone.  The  solid  angle  may  also  be  defined  as  the  ratio  of  the 
area  cut  out  upon  any  sphere  concentric  with  the  vertex  of  the  cone,  to  tlie  S(juare 
of  the  radius  of  the  sphere  (compare  the  definition  of  the  angle  between  two  lines 


348  INTEGRAL   CALCULUS 

in  radians).  Show  geometrically  (compare  Ex.  10,  p.  21*7)  that  the  infinitesimal  .solid 
angle  dw  of  the  cone  which  joins  the  origin  r  =  0  to  the  periphery  of  the  element  dS 
of  a  surface  is  cZw  =  cos(j',  n)dS/r-,  where  (r,  n)  is  the  angle  between  the  radius 
I)roduced  and  the  outward  normal  to  the  surface.    Hence  show 

r  cos  (;•,  ?()  ,  r  r-dS        rid)-,  r  d   1  ^  ,  T  ,,,  ,,  1 

0,  =  I    5__!  as  =       =  /   -     —  dS  =  -  I dS  =  -  I  dS'V  - , 

J         r2  J      r»        J    r-  dn  J   dn  r  J  r 

where  the  integrals  extend  over  a  surface,  is  the  solid  angle  subtended  at  the  origin 
by  that  surface.    Infer  further  that 

-f^ldS  =  4n     nr     -rAl,,.s  =  o     or     -f    '^^^^dS  =  d 

Jq  dn  >'  'JQ  dn  r  Jq  dn  r 

according  as  the  point  r  =  0  is  within  the  closed  surface  or  outside  it  or  upon  it 
at  a  point  where  the  tangent  planes  envelop  a  cone  of  solid  angle  ff  (usually  'Aw). 
Note  that  the  formula  may  be  applied  at  any  point  (|,  ??,  f)  if 

where  (,r,  ?/,  z)  is  a  point  of  the  surface. 

8.  Gaunti's  Intcgnd.  Suppose  that  at  r  =  0  there  is  a  particle  of  mass  in 
which  attracts  accnrding  to  the  Newtonian  Law  F  =  ni/r-.  Show  that  the 
])oteiitial  is  !'=—)»//•  so  that  F=— VI'.  The  induction  or  flux  (see  Ex.  10, 
p.  308)  of  the  force  F  outward  across  the  element  (ZS  of  a  surface  is  by  definition 
—  Fcos(F,  }t)dS  ■=  F'dS.  Show  that  the  total  induction  or  fiux  of  F  across  a 
surface  is  the  surface  inteu'ral 


fF.rZS  =:  -  fdS.YV  =  -  f  --  dS  =  m  fdS-V  ^ ; 
J  J  J     dn  J  r 

'1=-'    f  F.dS  =  ±   f  dS.V  V=-'    r  1  "^  dS, 
4  TT  Jq  4  TT  Jo  4  TT  Jq  '7h  r 


dn 

_  1     /•  ^      n 

and 


where  the  sui'face  integral  extends  ovei'  a  surface  surrounding  a  point  r  =  0,  is  the 
fornuila  for  obtaining  the  mass  in  within  the  surface  from  the  field  of  force  F 
whicli  is  .set  up  bj'  the  mass.  If  there  are  several  masses  ?/ij,  ?«.,,  •  •  •  situated  at 
points  (fj,  77j.  j-j),  (t._,,  r,.^,  f^),  ....  let 

F  =  Fi  +  F, +  ••••       r=  r,  + ]',  +  ..., 

be  the  force  and  poteiUial  at  {x,  y,  z)  due  to  the  masses.    Show  tliat 

-^-    C^.dS^    ^-    CdS.\V=-    ^    V    r    ''    ^  r7.s- =  V'/«f  =  .V,         (25) 
Att  J.J  ■\ttJz  -i  TT  jQ  J  j  dn  n  ^ 

where  li  extends  over  all  the  mass(\s  and  ^'  over  all  I  he  masses  within  the  surface 
(none  l)eing  on  it),  gives  the  total  mass  M  within  tht;  surface.  The  integral  (25) 
which  gives  the  mass  within  a  surface  as  a  surface  integral  is  known  as  Gauss's 
Integral.  If  the  foi'ce  wer«'  rei)ulsive  (as  in  electricity  and  magnetism)  instead  of 
attracting  (as  in  gravitation),  the  results  would  be   I'  =  nt/r  and 

-^-    rF.c/S  =  --i    r./S.Vl'  =  -'y    f    '^    '"'dS^yuu^M.  (25-) 

4  7rc/o  \TrJj  \tv  J^  J.jdn    r,-  ^ 


ox  MULTIPJ.E  INTEGRALS  349 

9.  IfV  =  i hj hk  —  be  the  operator  defined  on  page  172,  show 

ex        cy         cz 

cz       cy       cz  \cy       cz]         \cz       cs  ]  \cx       cy  / 

by  formal  operation  on  F  =  i'i  +  Qj  +  ii'k.    Sliow  further  that 

VxV  U=0,         V-VxF  =  0,  (VV)  (*)  =  (~  +  — ,  +  ^  (*), 

Vx(VxF)  =  V  (V'F)  -  (V'V)  F         (write  tlie  Cartesian  form). 
Show  that  (V'V)  U  =  V.(VL').    If  u  is  a  constant  nnit  vector,  show 

,^      cF  (F  (F  dF 

(U'V)  F  =  —  cos  <-<•  -I cos  j:i  -\ cos  y  =  — 

ex  cy  cz  (Is 

IS  the  directional  derivative  of  F  in  the  direction  u.    Show  (Jr'V)F  =  dF. 

10.  Green's  Formula  (space).  Let  F{x,  y,  z)  and  G  {x,  ?/,  z)  hi',  two  functions 
so  that  VF  and  VG  become  two  vector  functions  and  FVG  and  GVF  two  other 
vector  functions.    Show 

V.(FVG-')  =  Vi^.VG  +  FV.VG,         V.(GVF)  =  VF-Vr/  +  6T.VF, 

or  A/f^^U-^-(f^U-(f-) 

ex  \     cx /      cy\     cy/       cz\     cz/ 

cF  cG      cF  cG      cFcG       ^/r-G       c-G      c^G\ 
cx   cx       cy  (y       cz    cz  \cx-       cy-        cz^ I 

and  the  similar  expressions  which  are  the  Cartesian  ecpiivalents  of  the  above  vector 
forms.    Apply  Green's  Lennna  or  (iauss's  Formula  to  show 

r  FV  G'.rZS  =  r  VF.  V  Gd  V  +  f  ^•^'^- V  Gd  V,  (26) 

C  GVF.fZS  =  fvF.VGdV  +  fGV.VFdV,  (26') 

HfVG  -  G'VF).rZS  =  C{FV.VG  -  G\''VF)dV,  (26") 

Cr^dG^^,        r/cF  cG       cF  cG       cF  cG\  ^^^       r^Vf"'^       c^^       f^^\  ^t^ 
./Q     tin  «^  \cx   cx       cy   cy       cz    cz;  J      \fX^       cy        cz^ / 

Jo\     dn  dnj  J   y_    \dx-       cy-        cz'^ /  \cx'-       cy-        cz'^ / A 

The  formulas  (26),  (26'),  (26")  are  known  as  Green's  Formi(l(t.s;  in  particular  the  first 
two  are  asymmetric  and  the  third  synnnetric.  The  ordinary  Cartesian  forms  of 
(26)  and  (26")  are  given.  The  expression  c'-F/cx^  +  c'-F/cy^  +  c'^F/cz-  is  often 
written  as  AF  for  brevity  ;  the  vector  form  is  V-VF. 

11.  From  the  fact  that  the  integral  of  F.cZr  has  opposite  values  when  the  curve 
is  traced  in  opposite  directions,  show  that  the  integral  of  VxF  over  a  closed  surface 
vanishes  and  that  the  integral  of  V*VxF  over  a  volume  vanishes.  Infer  that 
V.VxF  =  0. 


350  INTEGRAL  CALCULUS 

12.  Reduce  the  integral  of  VxVf^^  over  any  (open)  surface  to  the  difference  in 
the  values  of  U  at  two  same  points  of  the  bounding  curve.   Hence  infer  VxVf  =  0. 

13.  Comment  on  the  remark  that  the  line  integral  of  a  vector,  integral  of  F«dr. 
is  around  a  curve  and  along  it,  whereas  the  surface  integral  of  a  vector,  integral 
of  F»dS,  is  over  a  surface  but  through  it.  Compare  Ex.  7  with  Ex.  ](?  of  p.  2'M.  In 
particular  give  vector  forms  of  the  integrals  in  Ex.  16.  p.  297,  analogous  to  those  of 
Ex.  7  by  using  as  the  element  of  the  curve  a  normal  dn  ei^ual  in  length  to  di, 
instead  of  dr. 

14.  If  in  F  =  Pi  +  Qj  +  /'k,  the  functions  P.  Q  depend  only  on  x.  y  and  the 
function  2?  =  0,  apply  Gauss's  Formula  to  a  cylinder  of  unit  height  upon  the 
x;/-plane  to  show  that 

CV'FdV  =  fF'dS     becomes     f  f  (~  +  ^)  dxdy  =  fF-dn, 

where  dn  has  the  meaning  given  in  Ex.  13.  Show  that  numericallj'  F»dn  and  FxrZr 
are  equal,  and  thus  obtain  Green's  Lemma  for  the  plane  (22)  as  a  special  case  of  (20). 
Derive  Green's  Formula  (Ex.  10)  for  the  plane. 

15.  If     CF'dr  =^  CC'dS,  show  that    /"(G  -  VxF).rZS  =  0.    Hence  infer  that  if 

these  relations  hold  for  every  surface  and  its  bounding  curve,  then  G  =  VxF. 
Ampere's  Law  states  that  the  integral  of  the  magnetic  force  H  about  any  circuit  is 
equal  to  Att  times  the  flux  of  the  electric  current  C  through  the  circuit,  that  is, 
through  any  svirface  spanning  the  circuit.  Faraday's  Law  states  that  the  integral 
of  the  electromotive  force  E  around  any  circuit  is  the  negative  of  the  time  rate 
of  flux  of  the  magnetic  induction  B  through  the  circuit.  Phrase  these  laws  as 
integrals  and  convert  into  the  form 

4  ttC  =  curl  H,  -  B  =  curl  E. 

16.  By  formal  expansion  prove  V'(ExH)  =  H'VxE  -  E'VxH.  Assume  VxE  =  -H 
and  VxH  =  E  and  establish  Povnting's  Theorem  that 


J(ExH).r?S  =  -  i-  J 1  (E.E  +  H.H) 
17.  The  "  e(iuation  of  continuity  "  for  fluid  motion  is 


dV 


h H '-  + ~  =  0     or     —  +  7;    --'+-"  +  _ ^    =  0, 

ct  cx  ly  cz  dt  \  cx        (1/        cz  / 

where  I)  is  tlie  density,  v  —  u,r  +  ji'„  +  ki\  is  the  velocity.  cD/ct  is  tlie  rate  of 
change  of  the  density  at  a  point,  and  dl)/dt  is  the  rate  of  eliaiige  of  density  as  one 
moves  witli  the  fluid  (Ex.  14.  p.  101).  Exiilain  the  meanini:-  of  the  e(iuation  in  view 
of  the  work  of  the  text.    Show  that  for  fluids  of  constant  density  V-v  =  0. 

18.  If  f  denotes  the  acceleration  of  the  particles  of  a  fluid,  and  if  F  is  the 
external  force  acting  per  unit  mass  upon  the  elenieius  of  fluid,  and  if  p  denotes 
the  pressure  in  the  fluid,  show  that  the  eciuation  of  motion  for  the  fluid  within  any 
surface  may  be  written  as 

V  i-Dd T  =  V F/AZP  -  V  pf^S     or      CiBd  1 '  =  CfImI  1 ' -  C pdS. 


ON  MULTIPLE  INTEGRALS  351 

where  the  suminatioiis  or  integrations  extend  over  the  vohime  or  its  bounding  sur- 
face and  tlie  pressures  (except  tliose  acting  on  the  bounding  surface  inward)  may 
be  disregarded.    (See  the  first  half  of  §  80.) 

19.  By  the  aid  of  Ex.  6  transform  the  surface  integral  in  Ex.  18  and  find 

fDidV  =  f{I)F  -  Vp) dV    or     —  =  F  -  -  Vp 
J  J  dt-  1) 

as  the  equations  of  motion  for  a  fluid,  where  r  is  the  vector  to  any  particle.   Prove 

,    ,  (7-r  dv  fV      ,  fv  1     ,       , 

dt-'  dt  ct  ct  2 

,    .    (^  ,  1      ■.  ,    '^v  '/r  ,     d-T      ]  , , 

dt  dt  dt  dt-       2 

20.  If  F  is  derivable  from  a  potential,  so  that  F  =  —  Vr,  and  if  the  density  is  a 
function  of  the  pressure,  so  that  dp/l>  =  dP,  show  that  thr  equations  of  motion  are 

£Z_  vxvxv=- V  (6^+ P  +  ^i-^),    or     -{v.dT}=-d(u+F-l^i---' 

after  nudtiplication  by  dr.  The  first  form  is  Hehnholtz"s,  the  .second  is  Kelvin's. 
Show 

/>  -'•.  'J,  z  d  dp  ^'  !''•  r  1    -.1  ■^'  ''• "  r 

I  —  (v.(?r)  =  —   I  \-di  =  —     L  +  P 1-2  and    |   V'dx  =  const. 

Jn,h,c      dt  dtJfi.h.c  L  2       J(7, '),  o  Jq 

In  particular  explain  that  as  the  differentiation  d/dt  follows  the  i)articles  in  their 
motion  (in  contrast  to  c/ct.  which  is  executed  at  a  single  point  of  space),  the 
integral  nuist  do  so  if  the  order  of  differentiation  and  integration  is  to  be  inter- 
changeable. Interpret  the  final  equation  as  .stating  that  the  circulation  in  a  curve 
which  moves  with  the  fluid  is  constant. 


21. 


If  ^  +  ^^  +  ^  =0,  show  C\m^  (-)V(-T>r=  f  u'l^ds. 

ex-        eg-        cz-  J  \_\czl       \cy  /      \  c2  /  J  Jq     dn 


22.  Show  that,  apart  from  the  proper  restrictions  as  to  contiiniity  and  differen- 
tiability, the  neces.sary  and  sufficient  condition  that  the  surface  integral 

C  C  Pdydz  4-  ({dzdx  +  Hdxdy  =  C  pdx  +  qdy  +  rdz 

depends  only  on  the  curve  bounding  the  svirface  is  that  P.^  +  (^,^  -f-  li'^  =  0.  Show 
further  that  in  tliis  case  tlie  surface  integral  reduces  to  the  line  integral  given  above, 
jjrovided  p,  17,  r  are  such  functions  that  r,^  —  7'  =  P.  }k  —  ?'^  =  Q.  q',.  —  p',,  —  P- 
Show  finally  that  these  differential  equations  inv  j>.  7.  r  may  be  .satisfied  by 

and  determine  by  inspection  alternative  values  of  p,  q,  r. 


CHAPTER  XIII 


ON  INFINITE  INTEGRALS 


140.  Convergence  and  divergence.    The  definite  integral,  and  hence 
for  theoretical  purposes  the  indefinite  integral,  has  been  defined, 

f  f{T)d,;  F(,r)=   r f(:r)ch; 

when  the  function  /"(*')  is  limited  in  the  interval  a  to  h,  or  «  to  cr ;  the 
proofs  of  various  propositions  have  depended  essentially  on  the  fact 
that  the  integrand  remnined  finite  over  the  finite  Interval  of  Integration 
(§§  16-17,  28-30).  Nevertheless  problems  which  call  for  the  determina- 
tion of  the  area  between  a  curve  and  its  asym})tote,  say  the  area  under 
the  witch  or  cissoid, 


L 


8  aN.<- 
x^  +  4  a? 


—  4  rr  tan" 


2  a 


=  4  7r«", 


— .  =  3  TTft", 

„      V2  a  —  X 


have  arisen  and  have  been  treated  as  a  matter  of  course.*    The  inte- 
grals of  this  sort  require  some  special  attention. 

When  the  Integrand  of  a  definite  Integral  heeomet^  Infinite  vlthln  or 
at  the  extremities  of  t/ie  Inti'riuit  of  Intcgrathm,  or  when  one  or  both  of 
the  limits  of  Integration  heeome  Infinite,  the  Integral  Is  ealled  an  infinite 
Integral  and  Is  defined,  not  as  the  limit  of  a,  sum,  hat  as  tlie  limit  of  an 
Integral  trlfh  a  variable  limit,  tliat  Is,  as  the  limit  of  a  funetlon.    Thus 


f{x)dx  =  lim 


f(x)dx=  lin 


U  a 


f(,r)dx 


f(x)dx 


infinite  u})per  limit, 


integrand  f(b^  =  co. 


These  definitions  may  l)e  illustrated  by  figures  which  sliow  the  coiniec- 
tion  with  the  idea  of  ai'ca  l)etween  a  curve  and  its  asymptote.  Similar 
(h'finitions  would  l)e  given  if  the  lower  limit  were  —  co  or  if  the  inte- 
grand became  infinite  at  ./•  =  ff.  If  the  integi'aiul  were  infinite  at  some 
intermediate  point  of  the  interval,  the  interval  would  be  subdivided 
into  two  intervals  and  tlie  definition  would  be  a])])lied  to  each  part. 


*  Here  and  below  the  eoustnu'tidii  of  figures  is  left  to  the  reader. 
352 


ox  INFINITE  INTEGRALS 


353 


Now  the  behavior  of  /•^(.r)  as  ;r  approaches  a  definite  value  or  lieooines 
infinite  may  be  of  three  distinct  sorts  ;  for  F(.r)  may  ai)])i'oach  a  definite 
finite  quantity,  or  it  may  l)eeome  infinite,  or  it  may  oscillate  without 
ap})roaching  any  finite  quantity  or  becoming  definitely  infinite.  The 
examples 


Jo     •*^'+4«- 

X 


cos  ,'/v/,> 


=  lim 


lim 


lim 


J  I      COf 
0 


=  4  crtan"^ 


X 


osav/.r  =  sin  ,r 


=  2  7r«",     a  limit, 
becomes  infinite,  no  limit, 
oscillates,  no  limit. 


illustrate  the  three  modes  of  behavior  in  the  case  of  an  infinite  up})er 
limit.  In  the  first  case,  where  the  limit  exists,  the  infinite  integral  is 
said  to  ermrerge;  in  the  other  two  cases,  Avliere  the  limit  does  not  exist, 
the  integral  is  said  to  direrge. 

If  the  indefinite  integral  can  be  found  as  above,  the  question  of  the 
convergence  or  divergence  of  an  infinite  integi'al  may  be  determined 
and  the  value  of  the  integral  may  be  obtained  in  tlu'  case  of  convergence. 
If  the  indefinite  integral  cannot  be  found,  it  is  of  prime  importance  to 
know  whether  the  definite  infinite  integral  converges  or  diverges ;  for 
there  is  little  use  trying  to  compute  the  value  of  the  integral  if  it  does 
not  converge.  As  the  infinite  limits  or  the  points  where  the  integrand 
becomes  infinite  are  the  essentials  in  the  discussion  of  infinite  integrals, 
the  integrals  Avill  be  Avritten  with  onlv  one  limit,  as 


j  /{■'■)  <^-r,  [/(■-)''■>;        jf 


(x)  dx. 


To  discuss  a  moi'e  complieatcd  (•()nd)inati()n,  one  would  Avrite 


J,,     V,/rMog,''     Jo       Jj       Ji       J- 


V.H( 


and  treat  all  four  of  the  infinite  inte"'rale 


Jr     er'dx  r^     e-'dx  r     e-^'dx  r"     e^'dx 

„  Va'''log.T  J     V.r^log.r  Jj  V.'''Uog.r  J       Vy'loga- 

Now  by  definition  a  function  A'(.r)  is  called  an  /-.'-function  in  the 
neighV)orhood  of  the  value  ,/•  =  n  when  the  function  is  continuous  in 
the  neighboi'hood  of  .'■  =  a  and  approaches  a  limit  Avliicli  is  neither  zero 
nor  infinite  (p.  Oli).     Tlie  hejiavinr  of  tlie  infinite  integrols  of  a  finvtion 


354  INTEGKAL   CALCULUS 

vldclx  does  not  change  sign  and  of  the  i>voduet  of  that  function  Jnj  on 
1-1  function  ore  identi^'al  as  for  as  convergence  or  divergence  ore  concerneit . 
Consider  the  proof  of  this  theorem  in  a  special  case,  namely, 

j     f{.r)d.r,  f  f  [,')]■:  (x)dx,  (1) 

where  f(^)  may  h(»  assumed  to  remain  positive  for  large  values  of  ,/• 
and  E{^')  approaches  a  positive  limit  as  ,/■  becomes  inhnite.  Then  if  A' 
be  taken  sufficiently  large,  both /(./•)  and  />'(.'•)  have  become  and  will 
remain  })ositive  and  finite.    By  the  Theorem  of  the  Mean  (Ex.  5,  p.  29) 


I  f  f(,r)d.r  <    f   f(,-)K(x)dx  <  M  f  f(,r)d:r, 

J  K  Jk  J  K 


X  >  K, 


where  ui  and  M  are  the  minimum  and  maximum  values  of  /..'(,/•)  l)etween 
K  and  co.  Xow  let  x  become  infinite.  As  the  integrands  are  positive, 
tlie  integrals  must  increase  with  x.    Hence  (p.  3o) 

if    I     /(.r)rf.c  converges,       I     f(x)E(x)(/x<MJ     f(x)(tx  converges, 
Jk  J  k  J  k 

if  I     /(,/■)  E  (,r)  dx  converges, 

Jf(x)dx  <  -—    \      t\x)E{x)dx  converges: 
K  ■  "'  Jk' 

and  divergence  may  be  treated  in  the  same  way.    Hence  the  integrals 

(1)  converge  or  diverge  together.    Tlie  same  treatment  could  be  given 

for  the  case  the  integrand  became  inhnite  and  for  all  the  variety  of 

hypotheses  Avhich  could  arise  under  the  tlicorem. 

Tliis  theorem  is  due  of  the  most  useful  ami  most  easily  applied  for  (letermiiiiu!; 
the  converirence  or  divergence  of  an  inlinite  integral  with  an  integrand  which 
does  not  change  sign.    Thus  consider  the  case 

Ilei'e  a  simple  rearrangement  of  the  integrand  throws  it  into  the  product  of  a  func- 
tion E{.r).  which  ajiju-oaches  the  limit  1  as  x  becomes  inlinite.  and  a  function  l/.r-. 
the  integration  nf  which  is  possible.  Hence  liy  the  thedrem  the  original  integral 
converges.  This  cnuld  have  been  seen  V)y  integrating  the  original  integral  :  l)ut 
the  integration  is  not  altogether  short.  Another  case,  in  which  the  integration  is 
not  possible,  is 

r^       dr        _   r^  1  ''■'■ 

•^     ^'l- j:^      -^     \'l  +  x^  v'l  +  j;  V 1  -  x ' 

1  ^1      dx  I |i 


UN  INFINITE  INTEGRALS  355 

Here  £(1)  =  h    The  iiite_i,a-al  is  again  convergent.    A  case  of  divergence  would  be 

C         lU              r         1         dx          -^ ,  ^             1  r  dx  2   I 

I    =  I >        E{x)  = ,  I    —  = 


(2x-/^)i      •^M2-/)ixi  (2-a:)t  -^Og.!  V, 


XlQ 


141.  The  interpretation  of  a  definite  integral  as  an  area  will  suggest 
another  form  of  test  for  convergence  or  divergence  in  case  the  inte- 
grand does  not  change  sign.  Consider  two  functions  fiy)  and  </'(.'-■) 
both  of  Avhich  are,  say,  positive  for  large  values  of  x  or  in  the  neigh- 
borhood of  a  value  of  ./•  for  which  they  become  infinite.  If  fltc  cur  re 
y  =  y(/{.r)  rfiiKt'ins  nhorr  y  =zf(.r),  ihe  intrfjrid  of f{.r^  uiust  (■Dnrrrtje  if 
t/ie  integral  of{f/(.r)  eonrcrf/es,  and  the  infegrnl  of{l/(.r)  iiuiat  dircrge  if 
the  integnd  <ff(-r)  dirm-gea.  This  may  be  proved  from  the  definition. 
For /(,/•)  <  \\i{x)  and 

j      f{.r)d.r  <    /      ip(x)dx       or       F(.r)   <  "^  (•'■). 

Jk  J  k 

Now  as  X  approaches  h  or  x,  the  functions  F(x)  and  '^(x)  both  increase. 
If  >!'('./•)  approaches  a  limit,  so  must  Fix)  ;  and  if  F(j')  increases  wdth- 
out  limit,  so  nnist  "^(x). 

As  the  relative  behavior  oif(x)  and  \p(x)  is  consequential  only  near 
particular  values  of  x  or  when  x  is  vcr}-  great,  the  conditions  may  be 
expressed  in  terms  of  limits,  namely  :  Jfipix)  does  not  ehungc  sign  and 
if  the  ratio  f(x)/\l/(x)  (ipiironclirn  a  finite  limit  {i>r  ':ero),  tJie  integral  of 
f(:>'')  icill  conrrrge  if  tin-  inti'gml  of  (//(,'■)  converges ;  and  if  the  ratio 
f(x)/\p(x)  (ipproacln's  a  jinitc  limit  (not  zer(t)  or  hccomcs  infnite,  the 
integral  iff(x)  mill  direrge  if  the  integral  of  \p(x)  dirergis.  For  in  the 
first  case  it  is  possible  to  take  x  so  near  its  limit  or  so  large,  as  the 
case  may  be,  that  the  ratio  /'(,'' )/'«/' (■'')  shall  be  less  than  any  assigned 
number  G  greater  than  its  limit;  then  the  functions /(./-j  and  G\p(x) 
satisfy  the  condititjns  estal)lished  above,  namely  /  <  G't/',  and  the  inte- 
gral of /(./•)  converges  if  that  of  !//(.'•)  does.  In  like  maimer  in  the  second 
case  it  is  possible  to  proceed  so  far  that  the  ratio /('./•) /i// (';/;)  shall  have 
become  to  remain  greater  than  any  assigned  nund)er  g  less  than  its 
limit;  then/'>  g\p.  and  tlie  result  abcn'e  may  be  ajjplied  to  show  that 
the  integral  of /'(,/•)  diverges  if  that  of  </'(.'•)  does. 

For  an  infinite  up})er  limit  a  direct  integration  shows  tliat 


f 


dx  _    -  1       1 


or  loi 


converges  if  /,•  >  1,    ^, 
diver.yes  if  /,•  ^  1. 


Now   if   the    /r.s-/  function    cf)(x)    be    chosen   as    1/x''  =  x~^,   the    ratio 
f(_x)/cj)(x)  Viecomes  x'''f{x).  and  if  tJif  limit  <f  tlie  i)roduct  J^f^J')  exists 


356  INTEGRAL  CALCULUS 

and  may  he  shown  to  he  finite  (or  zero)  as  x  becomes  infinite  for  any 
cJioice  of  k  greater  than  1,  tlie  integral  of  fix)  to  infinity  will  converge; 
hut  if  the  product  ajyproacltes  a  finite  limit  (not  zero)  or  hecomes  infinite 
for  any  choice  of  k  less  than  or  equal  to  1,  tlie  integral  diverges.  This 
may  be  stated  as  :  The  integral  oif(x)  to  infinity  will  converge  if /(a') 
is  an  infinitesimal  of  onler  higher  than  the  first  relative  to  1/x  as  x 
becomes  infinite,  but  will  diverge  if /(•'•)  is  an  infinitesimal  of  the  first 
or  lower  order.    In  like  manner 


f 


dx 


{h -.,■)''      k-1  (/y-./-)^- 


or  —  log(/;  —  x) 


converges  if  /.•<!, 
diver":es  if /.'^l,     ^ 


and  it  may  be  stated  that:  The  integral  of/(.x')  to  h  will  converge  if 
f{x)  is  an  infinite  of  order  less  than  the  first  relative  to  (f>  —  x)"^  as  x 
approaches  h,  but  Avill  diverge  if /(.r)  is  an  infinite  of  the  first  or  higher 
order.    The  proof  is  left  as  an  exercise.  See  also  Ex.  3  below. 

As  an  example,  let  the  integral    I     x"e-nU  be  tested  for  convergence  or  diver- 
"        J  0 
gence.    If  n  >  0,  the  integrand  never  becomes  infinite,  and  the  only  integral  to 

examine  is  that  to  infinity  ;  but  if  ?i  <  0  the  integral  from  0  has  also  to  be  consid- 
ered. Now  the  function  e-^  for  large  values  of  x  is  an  infinitesimal  of  infinite 
order,  that  is,  the  limit  of  x^'  +  "e-^'  is  zero  for  any  value  of  k  and  n.  Hence  the 
integrand  x"e-^  is  an  infinitesimal  of  order  higher  than  the  first  and  the  integral 
to  infinity  converges  under  all  circumstances.  Forx  =  0,  the  function  e--«  is  finite 
and  equal  to  1  ;  the  order  of  the  infinite  x"e-^  will  therefore  be  precisely  tlie  order 
71.  Hence  the  integral  from  0  converges  when  7i  >  —  1  and  diverges  when  ?i  g  —  1. 
Hence  the  function 

T(a)=f    x'^-ie-aJx,         a:  >  0, 

defined  by  the  integral  containing  the  parameter  (t,  will  be  defined  for  all  positive 
values  of  the  parameter,  but  not  for  negative  values  nor  for  0. 

Thus  far  tests  have  been  established  only  for  integrals  in  which  the 
integrand  does  not  change  sign.  There  is  a  general  test,  not  particularly 
useful  for  ])ractical  })urposes,  but  liighly  useful  in  obtaining  tlieoretical 
residts.    It  will  be  treated  nuu'elv  for  the  case  of  an  infinite  limit.    Let 


''■"'^f-' 


F  (■'•)=!     /(■'■) '^'^^,         F(,r")-F(x')=J       f{x)dx,         x',x">K.    (4) 

Xow  (Ex.  o,  ]).  44)  tlie  necessary  and  sufficient  condition  tliat  Fix) 
approach  a  limit  as  x  becoutes  inhnite  is  tliat  F(x")—F(x')  shall 
approach  tlic  limit  0  wlieii  x'  and  x",  regarded  as  independent  varia- 
bles, become  infinite;  liy  tlie  definition,  tlien,  tliis  is  the  necessary 
and  suffi(dent  condition  tliat  tlie  integral  of  ./'(•'")  ^'^  infinity  shall 
conversj-e.    Furthermore 


ON  INFINITE  INTEGRALS 


357 


(f 


/■ 


/'(.>•)  I  dx     conoerges 


,  then     I 


f(.r)dx 


(5) 


must  converge  ami  is  said  to  be  absolt(teIi/  conrei-r/ent.    The  proof  of  this 
iiuportant  theorem  is  contained  in  the  above  and  in 


£  /co^^-'-^j^  I /(•'•)  I '/•'•• 


To  see  whether  an  integral  is  absohitel}-  convergent,  the  tests  estab- 
lished for  the  convergence  of  an  integral  with  a  positive  integrand 
may  be  applied  to  the  integral  of  the  absolute  value,  or  some  oljvious 
direct  method  of  eomptiri^on  may  be  employed  ;  for  example, 


/cos  jilx        r '    1  (/,/•  . 

— ; ;  —  I      -T ,   which 
a-^x-     J      ./-  +  ..■- 


converges, 


and  it  therefore  appears  that  the  integral  on  the  left  converges  abso- 
lutely. When  the  convergence  is  r.ot  absolute,  the  question  of  con- 
vergence may  sometimes  be  settled  by  hitrijrdfioii  hi/  parts.  For 
suppose  that  the  integral  may  be   written  as 

f  f(x)dx  =   r  cf>(x)ipi^x)dx=    cf>(x)  filf(x)d.r     -  r  ct>\x)  f^,(,i-)dx' 

by  separating  the  integrand  into  two  factors  and  integrating  by  parts. 
Now  if,  when  ./•  l)ecomes  infinite,  each  of  the  right-liand  terms  approaches 
a  limit,  then 


/ 


f(u-)d., 


lim 


(f>(x)  j  ij/l^xjdx     —  lim    I     <^'(.'-)  I  \l/{x)dxdx, 


and  the  integral  of  f(x)  to  infinity  converges. 

,  1  -1      *i  (  r'"  xQo^sdx     ^^        /--^  x  I  cos  J I  (7 J 

As  an  example  consider  the  converc-oiice  01   ( Here  (      -- 

J        a-  -\-  X'  J         a-  +  X- 

does  not  appear  to  l.)e  converuent  ;  for,  apart  from  the  factor  [cos /|  wiiich  oscilhites 

between  0  and  1,  the  integrand  is  an  intinitesimal  of  only  the  first  cnxler  and  the 

integral  of  such  an  integrand  does  not  converge;  the  original  integral  is  therefore 

apparently  not  absolutely  c(jnvergent.    However,  an  integration  by  parts  gives 


/■^xcosxdx       j-snix  r'      r'^  x- —  a- 
= 1    —  I cos.mx, 
(/■-  +  x^        a-  +  /-;       'J     {x-  +  a-)' 

r-''  X-  —  (i-  ,         r-'dx 

cos  xdx  <   I 

J     {X-  +  a-)-  J      X- 


u-  +  X- 

Now  the  integral  on  the  right  is  seen  to  be  convergent  and,  in  fact,  absolutely 
convergent  as  .;■  ln'comes  infinite.  The  original  integral  therefore  nmst  approach 
a  limit  and  be  convergent  as  x  becomes  intinite. 


358  INTEGRAL  CALCULUS 

EXERCISES 

1.  Establish  the  convergence  or  divergence  of  these  infinite  integrals: 
,    .    C^       dx  /o     r°^      ^^dx  ,  .     r"^      x-dx 

^      xVl  +  x^  ^      {a-  +  x-)^  J      (a2  +  x-^)l 

(5)    I   x'^-i(l  —  x)^-'^dx   (to  have  an  infinite  integral,  a  nuist  be  less  than  1), 
-^  -'o  Vax-.f-  -^i    xVx^-1 

«/0l  —  /*  *^'J(1  X)3  i/OI  —  X 


(X)  r'~^^^;L=., /.-<!, /.;  =  i,         (m)  r\'l^Li:!f:czx,A-<i. 

2.  Point  out  the  peculiarities  which  make  these  integrals  infinite  integrals,  and 
test  the  integrals  for  convergence  or  divergence  : 

r  1   /          1  \"  /-  1    ]i)"-X 

(a)           log-    dx,  conv.  if  ?i  >  -  1.  div.  if  n  s  _  l,  (^)    /         ~^dx, 

J  0    \       x/  J  0    \  —  X 

(7)    I     (— log.r)«(7j-,                  (5)    I   '  logsinjJj-,  (e)    |     /  li)gsin,r(7j;, 

(r).  r^..g(.  +  ^) ^^,  („  fv   f   -.  (^)  rv-(h>g^)V 

J             \         x/  1  +  x-               Jo   (smx  +  cosx)-*^^  Ju        \       .f/ 

r'^   -  r^               TTX 

(k)    I      x^(/,f,  (\)    I     IniTxtan — dx, 

Jo  Jo                       2 

-'-OO  Jo          (1      +     X)- 

/-i  logxfZx  ,  ,  r"  -('-",)' 

-'"  ^  1  -X-  "^'^ 

(i-)    /       ^-     '^^-^->dx.      ix)  e--'-cosh^xrix. 

I/O           1  +  b-x-  Jo 


(') 

/; 

e-^(/x 

\  xlog(x  +  1) 

(m) 

/; 

1   +  X 

(T) 

/; 

'  sin'-x  , 

dx, 

X- 

(T) 

/; 

'  x«-ilo£rx  , 

^ —  ax, 

1  +x 

3.  Pfiint  out  the  similarities  and  differences  of  the  method  of  ^-functions  and 
of  test  functions.  Compare  also  with  the  work  of  tliis  section  the  remark  tiiat  tlic 
detenninatiou  of  the  order  of  an  infinitesimal  or  infinite  is  a  i)rol)it'ni  in  indeter- 
minate forms  (p.  (Jo).  State  also  wliether  it  is  necessary  that /(x)/-/' (x)  or  x^/(x) 
should  a])iiroacli  a  linnt.  or  whetiier  it  is  sufficient  that  the  (nuiutity  remain  tiniti'. 
Distinguish  "of  order  liiglier'"  (p.  3o(J)  from  "of  higher  order"'  (p.  03);  see  Kx.  iS.  p,  Otj. 

4.  Discuss  the  convergence  of  these  integrals  and  prove  the  convergence  is 
absolute  in  all  cases  where  possible  : 

--^^-dx.  (p)  J     cosx2.7x,  (7)/  --dx, 

(5)    p-'^^^i'l^^,  (,,    f\-u^^.o,l3xdx,  (n    f\l"^+^dx, 

Jo  X  Jo  Jo       \  X-^ 


ox  INFINITE  INTEGRALS  359 

(tj)    I      (ix,  (^)    I     e-<^cos?;xdx,  (t)    I       — ^  dx, 

J  0     x'^  +  ^-  «/  0  Jo       ^_p 

,     ,      /''•^  ,  a  .  ,,,       /^«=  sill  X  COS  (XX    , 

(k)    I     x'^-%--^<=o^Pco.s(xsiii/3)dx,  (X)    I      tZx, 

Jo  Jo  X 

(m)    /     COS x2  COS 2  axdx,         (i/)    j     sin  ( 1 )dx,      (o)    f     ^ dx. 

Jo  Jo  \2       2  XV  '^o       x"^ 

5.  If  /i(x)  and  f„{x)  are  two  limited  functions  integrable  (in  the  sense  of 
§§  28-30)  over  the  integral  a^x^h^  show  that  their  product /(x)  =/^(x)/2(x) 
is  integrable  over  the  interval.  Note  tJiat  in  any  interval  5,-,  the  relations 
m-iima  ^  nn  s  3/,-  ^  J/i,J/o;  and  M-nM-u  —  mum^i  =  MiiM2i  —  Miim.2i  + 
Miim-2i  —  mijm-2i  =  ^hiOa  +  rii^iOn  hold.    Show  further  that 

£ /i(x)/,(x)  dx  =  liin  ^  /,(^^)/,(,t,)5,- 

=  lini  2  /,(e,)  r£'''  ^  V,(x)  (/x  -  £"  +  '{./;(fO  -f,{x)  dx}! , 

or  f''f{x)dx  =  Um^J\i^,)£''  +  'fJx)dx 

i 

=  lim  ^ /; a,)  r  r  /^(x)  jx  -  r '  /,(x)  czxl , 

or  J /(x)tZx  =/,(ij)  J/,(x)  Jx  +  lim  ^  [/,(,^)  -/,(s^-_i)]   £ /,(x)(7x. 

6.  TAe  Second  Theorem  of  the  Mean.  If /(x)  and  0(x)  are  two  limited  functions 
integrable  in  the  interval  «  ^  x  ^  h^  and  if  0  (x)  is  positive,  nondecrea.sing,  and 
less  than  7f ,  then 

r  V  ('C)/(-r)  dx  =  K  f  /(x)  dx,         a  ^  f  ^  'a 

And,   more   generally,   if  <p{x)   satisfies   —  co  <  i  ^  0  (x)  ^  A' <  cc  and   is  either 
nondecreasing  or  nonincreasing  throughout  the  interval,  then 

f  0  (x)/(x)  dx  =  k  f  f{x)  dx  +■  K  f  f{x)  dx,         a  ^  J  ^  h. 

In  the  first  case  the  proof  follows  from  Ex.  o  by  noting  that  the  integral  of 
0  (x)/(x)  may  be  regarded  as  the  limit  of  tlie  sum 

<t>  (?i)/ "/(x) dx  +  ^  [0(t^■)  -  0  {^i _ i)]  r  /(x)  (Zx  +  [jr  -  0  (fe,)]  J  /(x)  tzx, 

where  the  restrictions  on  0  (x)  make  the  coefficients  of  the  integrals  all  positive  or 
zero,  and  where  the  sum  may  consequently  be  written  as 

M  [0  (>S)  +  <P  (lo)  -  4>  (sS)  +  •  •  •  +  0  (^„)  -  0  (f;,  _  i)  +  iv  -  0  (L)]  =  /"A' 
if  /u  be  a  properly  chosen  mean  value  of  the  integrals  which  multiply  tliese  coeffi- 

cients  :  as  the  integrals  are  of  the  form  I    /(x)  cZx  where  ^  =:  «.  x, .  •  •  • ,  x„,  it  follows 

J^ 


300 


INTEGRAL   CALCULUS 


tliat  (U  must  be  of  the  same  form  where  a  =  f  =  h.    The  second  form  of  the  theorem 
follows  by  considering  the  function  ^  —  A-  or  A-  —  (j>. 

7.  If  <^(x)  is  a  function  varying  always  in  the  same  sense  and  approaching  a 
finite  limit  as  x  becomes  infinite,   the   integral   I     <p{x)f{x)dx  will  converge  if 

I     /(x)(lx  converges.    Consider 

f"'  4,{x)f{x)ax  =  ,p{x')J^f{x)dx  +  <p{x")j"'  f{x}dx. 

8.  If  <p{x)  is  a  function  varying  always  in  the  same  sense  and  approaching  0  as 
a  limit  when  x  =  oo,  and  if  the  integral  F{x)  of /(x)  remains  finite  when  x  =  cc, 

then  the  integral  |      (p{x)f{x)dx  is  convergent.    Consider 

^%{x)/(x)tZx  =  .^(x')  [F(f)  -  F{x')]  +  cp{x")  [F(x")-  F(.')]. 

This  test  is  very  useful  in  practice  ;  for  many  integrals  are  of  the  f  i  )rm  /      0  (x)  sin  x(/x 

where  <p{x)  constantly  decreases  or  increases  toward  the  limit  0  when  x  =  x;  all 
these  integrals  converge. 

142.  The  evaluation  of  infinite  integrals.  Afttn-  an  infinite  integral 
lias  been  proved  to  converge,  the  jirolilmu  of  calculating  its  value  still 
remains.  Xo  general  method  is  to  l)e  had,  and  for  each  integral  some 
si>ecial  device  has  to  be  discovered  whicli  will  lead  to  the  desired 
residt.  Ill  Is  iiiinj  fre(iuently  hr  ((i'C(niij>Jis]ic)l  hi/  cliooshuj  c  finictuu} 
F(z)  of  till'  coiiipli'x  rur'wJili'  r:  =  ./•  -J-  ///  n ml  infi(jr(itin(j  tlu'  finirfinn 
11  round  some  dosed  path  In  tin'  z-phinr.  It  is  known  that  if  the  points 
where  F(,-;)  =  A(a-,  ?/)  +  n'(.>',  _y)  ceast-s  to  have  a  derivative  F'(',v"). 
that  is,  wdiere  A' (a-,  ?/)  and  }'(,'■,  //)  ceaso  to  liave  cfjiitinuous  first  par- 
tial derivatives  satisfying  the  relations  A','  =  \"„  and  A',^  =  —  l'^-,  ^I'e  cut 
out  of  the  ])lane,  the  integral  of  F{::)  around 
any  closed  i)ath  wdiich  does  not  include  any  of 
the  excised  points  is  zero  (§  VIX).  It  is  some- 
times possible  to  select  such  a  function  i"(.v) 
and  such  a  ]»at]i  of  integration  that  part  of 
the  integial  of  tlie  complex  function  I'cduces 
to  the  given  infinite  integral  Avhile  the  rest  of 
the  intcgi-al  of  the  com})lex  function  may  l)e  com})utcd.  Thus  there 
arises  an  equation  wdiich  determines  tlu'  value  of  the  infinite  iutey-ral. 


-A^iB 


A  +  \B 


dz=-Vdx 
dz=->ridij     dz=id)j 


1 


d.~=<t,v 


-A 


O 


A 


Consider  the  integral 


/; 


X 

(•    wllJL 

-  »>  e'>  - 

-  e-  '-^ 

ell  is  known  to  conven 


•'0       X  .'o  2 /x  J II     2  ix      Jo      2  ix 


Xiiw 


dx 


suggests  at  once  tiiat  the  I'liiirtinn  r''-/z  he  exaiiiiiicil.    This  fuiictiiiii  has  a  definite 
derivative  at  every  puint  except  ^  =  0,  and  the  origin  is  therefore  the  only  jmint 


ox  INFINITE  INTEGRALS  361 

which  has  to  be  cut  out  of  the  phiiie.    The  integral  of  e~/z  around  any  path  such 
as  tliat  marked  in  the  figure  *  is  therefore  zero.    Tlien  if  a  Is  small  and  ^1  is  large, 

0=1    —dz=\       —  tZx  +  I idy  +  j         (U 

Jq,    Z  Ja         X  Jo      ^l   +  hj  J  A         X  +   ill 

p  0    (;-  iA  -II  p-a  (.h;  p+a  (.iz 

+   I    ^ r-  i<il/+  —  dx  +  /         —  dz. 

J/i   —  A   +   11/  J- A      X  J --a       Z 

r""^.zx  =  -r"^*^dx  =  -r-'^c^  and  r""^j^=r"i+^j.: 

J-.l      X  J -a        X  J  a  X  J-a       Z  J  -  a  Z 


But 


the  first  by  the  ordinary  rules  of  integration  and  the  second  by  Maclaurin"s 
Formula.    Hence 

Xpiz                  p  A   fiix g—  ix           /i  +  a  (J^ 
^—  dz  =   i       hi         h  four  other  integrals. 
J     Z                 Ja                  X     ■               J-  a       Z 

It  will  now  be  shown  that  by  taking  the  rectangle  sufficiently  large  and  the 
semicircle  about  the  origin  sufficiently  small  each  of  the  four  integrals  may  be 
made  as  small  as  desired.  The  method  is  to  replace  each  integral  by  a  larger  one 
which  may  be  evaluated. 

I  I     idy  '  ^  I      ' ' \t\dy  <  I      ~e~ I'dy  <  —  ■ 

I  Jo    A  +  hj        \~Jo     \A  +  i>j\'    '    -^      Jo    .i  A 

The.se  changes  involve  the  facts  that  the  integral  of  the  absolute  value  is  as  great 
as  the  absolute  value  of  the  integral  and  that  e'-^  -  '■>  =  e''-'t:^  ''.  \  e'-^  ]  =  1.  \A  +  'ii/\>  A . 
e- '.'<!.  For  the  relations  j  f'-' j  =  1  and  \A-\-iyl>^l,  the  interpretation  of  the 
quantities  as  vectors  suffices  (§§  ll-'-i)  ;  that  the  integral  of  the  absolute  value  is 
as  great  as  the  absolute  value  of  the  integral  follows  from  the  .same  fact  for  a  sum 
(p.  154).  The  absolute  value  of  a  fraction  is  enlarged  if  that  of  its  numerator  is 
enlarged  or  that  of  its  denominator  diminished.    In  a  similar  manner 

A  I    r'>  c-'-'-!'    .,    I       B 

a' 


Furthermore 


I  dx    <  dx  =  ^c-  '■    -  ,         I        idy\<- 

Ja  X  +   IB  J-A      Ji  B  'J;;-A  +  iy  \         J 

,        -  a       Z  'J  —  a  \Z  \  «^0 

r  +  "  dz       c'^  re^'ldtb 

J-  a       Z  Jtt  rt'" 

Xe'^             r-'       sin.c  B  A 

—  dz=         2i- dx-TTi  +  K.         'R|  <2— +  2e--B_  +  7re, 
^    Z                  Ja                   X  A  B 


where  e  is  the  greatest  value  of  |7;|  on  the  semicircle.  Now  let  the  rectangle  be 
so  chosen  that  A  =  Be^  ^  ;  then  \E\<zie~-^^  +  ire.  By  taking  B  sufficiently  large 
e'^2'^  may  be  made  as  small  as  desired;  and  by  taking  the  .semicircle  sufficiently 

*  It  is  also  possible  to  integrate  almig  a  semicircle  from  A  to  —  A,  or  to  coine  back 
directly  from  IB  to  the  origin  and  separate  real  from  imaginary  parts.  These  variations 
iu  niethdd  niav  be  left  as  exercises. 


362  INTEGKAL  CALCULUS 

small,  e  may  be  made  as  small  as  desired.  This  amounts  to  saying  that,  for  A  suffi- 
ciently large  and  for  a  sufficiently  small,  R  is  negligible.   In  other  words,  by  taking 

X'*^  sill  X  TT 

'- may  be  made  to  differ  from   —  by 
X  2 

as  little  as  desired.  As  the  integral  from  z.ero  to  infinity  converges  and  may  be 
regarded  as  the  limit  of  the  integral  from  a  to  A  (is  so  defined,  in  fact),  the  integral 
from  zero  to  infinity  must  also  differ  from  I  ir  by  as  little  as  desired.  But  if  two 
constants  differ  from  each  other  by  as  little  as  desired,  they  nuist  be  equal.    Hence 


«/()  X.  A 


As  a  second  exami)le  consider  what  may  be  had  by  integrating  e'^/{z-  +  k^)  over 
an  appropriate  path.    The  denominator  will  vanish  when  z  =  ±  ik  and  there  are 
two  points  to  exclude  in  the  z-plane.    Let  the  integral 
be  extended  over  the  closed  path  as  indicated.  .There  is 
no  need  of  integrating  back  and  forth  along  the  double 
line  Oa,  because  the  function  takes  on  the  same  values         /  Xd  z=ik 

and  the  integrals  destroy  each  other.    Along  the  large        I ^|; 

semicircle  z  =  Ee^'t'  and  dz  =  Bie^<i>d(j>.    Moreover  —R  O 

r°     e'^fZx  r"^  e"'Zx         r^  e-'^dx         ,       ,  , 

I       =  —  I =  I by  elementary  rules. 

J-  R  x"  +  k:^  Jo       a;2  +  k'^      Jo    x^  +  k^ 

Hence        /       +  |     =  /       — -^^ dx  =  12  / dx, 

J-  R  x^  +  k"      Jo    X-  +  k-      Jo       x^  +  &'-  ^1,1    x~  +  k- 

and      0=  / dz  =  -2   (     dx  +   | ~+   /        

-'O  Z^  +  k-  Jo     X-  +  k-  Jo         R-e~  '*  +  k-  Jaa'a  z-  +  k- 

J^Tqw  le'^*'*!  =  •^ciR(eoi,<l>  +  iuin<t,)\^  =  |  (;- «sin  ,J,(^.,-«  cos  <^  |  =  (,- «  sin  </._ 

Moreover  |7i'^t''-^'*  +  k^\  cannot  possibly  exceed  A'-  —  k'^  and  can  e(iual  it  only  when 
(p  =  Itt.    Hence 

I     pn  c'''''"^Ric''l'dd)\  C'^   7.'e- /?  Bin  </,  ^^   ll(,-R%\ni, 

I  Jo      ifV'-^  +  A,-^  I      Jo       R--k-  Jo       R--Ic~ 

Now  by  Ex.  28,  p.  11,  sin  0  >  2  ^/tt.    Ilencc  tlio  integral  may  be  further  increased. 

I    r-  '''^■'^RU:>"d.p\  ^rl   Rr    '[  -jhp^        tt  _  ,  ,^,,_  -,. 
I  Jo       y.'-c-' '■'.''  +  A--  I   "     Jo         R-  -  A:-  R-  -  k-  '' 


Moreover,       (        -'  - '"    =  C      — ' '-~-    =  f      ('- h  v) - 

Jaa'a   Z-  +  k'^         Jaa'a   Z  +    ik  Z  -   //,•         Jaa'a  Vlkl  ' 


where  tj  is  nnifornily  iiiliiiitcsimal  with  the  radius  of  tlie  small  circle.  But 

Xdz                     .            -.      r         (''dz              2Trc~^ 
-  —  —  -Itvu     and  —  —   Y  t, 

.^a'a  Z-ik  Jaa'a   Z-  +   k-  2  k 

whei'e  '  j-|  ^2Tr€  if  e  is  the  lai'gest  value  of  |  7;  |.    Hence  tinally 


ox   IXFIXITE  IXTECrRALS 


363 


J'^  ''    CI  IS  X 
0    X-  +  k- 


-  t'" 
A; 


+  i-H ^ (t"''-  1). 


By  taking  the  siiuiU  circle  .small  enough  and  the  large  circle  large  enough,  the  last 
two  terms  may  be  made  as  near  zero  as  desired.    Hence 


x 


0     x^  +  Ic^ 


dx 


•2k 


(") 


It  may  be  noted  that,  by  tlie  work  of  § 


§  12*5.   f 

'J  aa'a 


dz 


is  exact 


z  +  kl  z  —  ki  2  ki 

and  not  merely  approximate,  and  remains  exact  for  any  closed  curve  about  z  ~  ki 
which  does  not  include  2:  =—  ki.  That  it  is  approximate  in  the  small  circle  follows 
innnediately  from  the  continuity  of  e'V(~  +  ^'O  =  e-^/2ki  +  -q  and  a  direct  inte- 
gration about  the  circle. 

As  a  third  example  of  the  method  let  1       -^ dx  be  evaluated.    This  inteirral 

Jo  1  +  x 
will  converge  if  0  <  a  <  1,  because  the  intinity  at  tlie  origin  is  then  of  order  le.ss 
than  the  first  and  the  integrand  is  an  infinitesi- 
mal of  order  higher  than  the  first  for  large  values 
of  X.  The  function  z"~^/{l  +  z)  becomes  infinite 
at  z  =  0  and  z  =—  1,  and  these  points  must  be 
excluded.  The  path  marked  in  the  figure  is  a 
closed  path  which  does  not  contain  them.  Now 
here  the  integral  back  and  forth  along  the  line 
aA  cannot  be  neglected ;  for  the  function  has  a 
fractional  or  irrational  power  z'^-i  in  the  nu- 
merator and  is  therefore  not  single  valued.  In 
fact,  when  z  is  given,  tlie  function  z"~'^  is  deter- 
mined as  far  as  its  absolute  value  is  concerned,  but  its  angle  may  take  on  any 
addition  of  the  form  2  7r/i-(ci:  —  1)  with  k  integral.  Whatever  value  of  the  function 
is  assumed  at  one  point  of  the  path,  the  values  at  the  other  points  mu.st  be  .such 
as  to  piece  on  contiimously  when  the  path  is  followed.  Thus  the  values  along  the 
line  aA  outward  will  differ  by  2  7r(cr  — 1)  fromtlio.se  along  ^Irt  inward  becau.se 
the  turn  has  been  made  about  the  origin  and  the  angle  of  z  has  increa.sed  by  2ir. 
The  double  line  be  and  cb.  however,  may  be  disregarded  because  no  turn  about  the 
origin  is  made  in  describing  cdc.    Hence,  remembering  that  c^'  =—1, 


Now 


0=         ~ dz=    f d(rt^^")=    (       dr+  ~ 

JqI  +  z  Jo      l-fre*'  J„     1  4- »•  J<>      1 

—^ e-^^'dr+   (       ^^ dz  +   I       ~     -dz. 

A  I  +   rC-'''  Jahha  1  +  Z  Jr.lc  1  -{-  Z 

/(I  yii  —lf^''2nai                  r*  A    i.<r  —  1 
dr  =         (1  -  c-^''')dr, 
A       1+  r              Ja     1  +  /• 


^.[a(,a4,i 


+  Acf 


id(p 


Ja     1  +  r 


I  Jo       l  +  Ac't'i        \      Jo      A-l\        1  .1-1 

I (Zz    ='    I     "?0i^  I       ( 

\Jahh,i  I  +  z  ;  J:; 77  1  4-  a(ft>'        i      Jo      I  —  a 


2  TTd" 


1-a 


564 


IXTE(  J  Pv AL   CALCULUS 

dz 


../c-1  +  2  J  1  +  2 

Hence     0  =  (1  -  t2'^<")  /       dr  +  ^Tric-"' +  i'.         \d< + 

Ja      I  +  r  ^1  —  1       1  —  a 


2  Tric'^'K 
2  7rA"       -iTrn" 


If  A  be  taken  sufficiently  l:\rge  and  a  .sufficiently  small,  j'  may  be  made  as  small 
as  desired.    Then  by  the  same  reasoning  as  before  it  follows  that 


0  =  (1  —  c-"'^')  I       fZr  +  2  TTic'^'",     o 


r     0 


sin  TTtr  I      dr  +  tt. 

Jo     1  +  r 


ami 


0       1 


fZjr,  = 

+  X  sin  air 


(8) 


143.   ( )iie  intt'gval  of  particular  iin})ortaiici'  is  I      e'^'dx.    Tlie  evalu- 
ation mav  lie  inadc  1)V  a  device  wliich  is  rarclv  useful.    Write 


r'.--.v.4r'.-.v.,.r'^-..vJ-=fr'r'. 


(l.rd  If 


The  })assage  from  tin;  product  of  two  iiitegrals  to  the  douljle  integral 
}nay  be  made  because  neither  the  limits  nor  the  integrands  of  either 
integral  depend  on  the  varial)le  in  the  other.  Now  transform  to  polar 
coordinates  and  integrate  over  a  quadrant  of  radius  ,4. 


,lnh,  = 


•'        ,  1 

c"-rdr<ie  +  R  =  77r(l 


• )  +  /', 


■\vliere  11  denotes  the  integral  o\'er  the  area  between  the  quadrant  and 
s(piare,  an  area  less  than  .V-l"  over  which  i'~'"^i~"^'.    Then 


11  <  iA-e--'\ 


n 


-•■'-, I. ,■<!, 


<\A\ 


Xow  .1  may  be  taken  so  large  that  the  (hmble  integral  differs  from  ^tt 
by  as  little  as  desii'ed.  and  hence  for  sufficiently  large  values  of  ,1  the 
simple  integral  will  differ  from  4-  Vtt  l)y  as  little  as  desired.    Hence  * 


i 


(-■' '<!.,'  =  1,  Vtt. 


(9) 


*  It  slicmld  lie  iidticcd  tlmt  tlif  pninf  just  ;^i\cii  dix's  imt  require  tlie  tlieory  of  intinitp 
dciulile  integrals  ihu-  of  rlian^e  ni  xarialde;  llie  wlmli'  proof  eousists  merely  in  tindini; 
a  nuiulier  \  Vn-  from  Avliieli  the  integral  may  lie  sliown  tn  dilfer  by  as  little  as  desired. 
'I'liis  was  also  true  of  the  jiroofs  in  §  14l':  no  tlieoi->-  had  to  lie  devcdoin-d  and  no  linntiiiu: 
;n'ocesses  were  used.  In  fact  tlie  exaluations  that  have  been  iii^rformed  show  i>(  tlieni- 
si'h'es  that  the  inliinte  inteurals  converLre.  F<ir  w  hen  it  has  heeu  sIkiwu  that  an  intejrral 
with  a  larire  enonudi  niii);-r  liiuit  and  a  small  enouirli  lnwer  limit  can  he  nnide  to  differ 
from  a  rertain  constant  by  as  little  as  desired,  it  has  thereby  l)een  proved  that  that 
integral  from  zero  to  inlinitv  must  roiiverLre  to  the  value  of  that  constant. 


ox  INFINITE  INTEGRALS  365 

When  some  infinite  integrals  have  been  evahiated,  otliers  niay  be 
obtained  from  them  In*  various  operations,  sucli  as  integration  by  parts 
and  change  of  variable.  It  should,  howevei-,  be  borne  in  mind  that  the 
rules  for  operating  with  definite  integrals  were  established  only  for 
finite  integrals  and  must  be  reestahl'islied  for  infinite  integrals.  From 
the  direct  application  of  the  definition  it  follows  that  the  integral  of 
a  function  times  a  constant  is  the  product  of  the  constant  by  the 
integral  of  the  function,  and  that  the  sum  of  the  integrals  of  two 
functions  taken  between  the  same  limits  is  the  integral  of  the  sum 
of  the  functions.  But  it  cannot  be  inferred  conversely  that  an  integral 
may  be  resolved  into  a  sum  as 

lf{x)  +  «/,  (a;)] dx  =   /    /'(.'•)  '/■'•  +  /     </>  (/'■;  dx 

when  one  of  the  limits  is  infinite  or  one  of  the  functions  becomes 
infinite  in  the  interval.  For,  tlie  fact  that  the  integral  on  the  left 
converges  is  no  guarantee  that  either  integral  upon  tlie  right  will 
converge;  all  that  can  be  stated  is  that  If  on<'  nf  iJic  Infcji-ah  on  the 
rhiht  cnnrpvgi's,  tlie  oilier  v'lU,  and  the  equation  Avill  Ijl'  true.  The 
same  remark  applies  to  integration  by  parts, 


X 


f(x)<f>'(x)dx  = 


/(,'-)  i>(-'-) 


f  (x)  </>  ( ./■;  dj 


If,  in  the  process  of  taking  the  limit  Avhich  is  required  in  the  defi- 
nition of  infinite  integrals,  tn-n  of  the  three  tmns  i/i  the  e'lUfitlon 
npproiirJt  /ii/iits,  the  third  irtll  (ipproaehi  a  limit,  and  the  efpiation  will 
be  true  for  the  infinite  integi'als. 

The  formula  for  the  chantfe  of  variable  is 


I 


"    \f(x)dx=   f  f[ci.(t)]4>'(t)df. 


where  it  is  assumed  that  the  derivative  ^'f)  is  continuous  and  does 
not  vanish  in  the  interval  from  t  to  T  (although  either  of  these  con- 
ditions may  V)e  violated  at  tlie  extremities  of  the  interval).  As  these 
two  quantities  are  equal,  they  will  a})proa('h  etjual  limits,  provided 
they  approach  limits  at  all,  Avhen  the  limit 

'    /(.r)rAv=    C\t\<i>(f)^^'(f)dt 

required  in  the  definition  of  an  infinite  integral  is  taken,  wliere  one  of 
the  four  limits  -'/,  h.  t^,  t,  is  infinite  or  one  of  the  integrands  becomes 


£ 


36G  INTEGKAL  CALCULUS 

infinite  at  the  extremity  of  the  interval.  Tli>;  fornnihi  for  the  rhanrje 
(if  mridhle  is  therefore  npplifnlili:  to  hijhiite  inte;/r'i.ls.  It  should  be 
noted  that  the  proof  applies  only  to  infinite  limits  and  infinite  values 
of  the  integrand  at  the  extremities  of  the  interval  of  integration  ;  in 
ease  the  integrand  becomes  infinite  Avithin  the  interval,  the  change  of 
variable  should  be  examined  in  each  subinterval  just  as  the  question 
of  convergence  was  examined. 

As  an  example  of  the  clianire  of  variable  consider  ( '-  dx  =   ~  and  takex  =  ax'. 

^  ^  Jo       X  -2 

X''="^  sin  (TX'  ,  ,        r  +  '^fi'max'  ,  ,               r~^  sino-x'  ,  ,           r-^'  =  '=  sin  ax'  ,  , 
dx  =  I         dx    or  =1  -      — dx  =—  I  dx  . 
„.  =  0            X'                     Jj'  =  0        X'                                J.r'  =  0         X'                           Jx'  =  0            x' 

according  as  a  is  positive  or  negative.    Hence  the  results 

/"'-    SinO-X    ,  ,     TT        .-  rv  1  "■        T  r^  /,«\ 

j dx  =  +         if     rt  >  0     and     if     a  <  0.  (10) 

Jo         X  2  2 

Sometimes  changes  of  vavialile  or  integrations  by  parts  will  lead  luick  to  a  given 
integral  in  such  a  way  tliat  its  value  may  be  found.    For  instance  take 

n  IT 

1=1  "  log  sin  ,/v/,c  :--- —   I      log- CDS //'/// =   I   "  Ing  ciis^d//.         y  =  - x. 

Then  21=  |"  "  (Ing.-iux  +  log  c(.sx)r/x  =  |"  - 1  ng -'-1:^  dx 

1  r~  TT  f  ■-•  TT 

=   -    I     lousiiixdx \i><z'2  =  I   ■  Ini.'- sin  xcZx log2. 

2  Ju       ■  2  J'>  2 

Hence  ^  =  f  ~ '"-  ""'" '^''''^  ^  ~'\>  ^^-'^'^-  (^^) 

Here  the  first  change  was  y  =  ■  tt  —  x.  The  new  integral  and  tlie  original  one 
were  then  added  togethci'  (tlie  variable  indicated  under  the  sign  of  a  definite  inte- 
gral is  immaterial.  ]>.  2(1).  and  tlie  sum  led  back  tn  the  original  integral  by  virtue 
of  the  substitution  //  =  2x  and  the  fact  that  the  curve  >/  =  log  sin  x  is  synnnetrical 
with  respect  to  x  =  J  tt.    This  gave  an  equation  winch  could  be  solved  for  I. 

EXERCISES 

It.  .  ^(''^  i-  ,  ,■       -  ,  r  '^    XSinX      ,  TT  , 

1.  Intesrate •  as  tor  the  case  of  (<).  to  show    |      dx  =    -  e-^'. 

2-  +  /.•-  .'o    X-  +  k-^  2 

2.  By  direct  integration  show  that    (     c~  ("-'"') -d^  converges  to  {a  —  ''0^^  when 
a  >  0  and  the  integral  is  exteude(l  along  the  line  >/  =  0.    Thus  prove  the  relations 

\     t-  "■''  COS  hxdx  = . ,  j     e-  "-^  sni  hxdx  = ,        a  >  0. 

•-'0  ('-  +  h-  Jo  (I-  +  h- 

AloiiLT  what  lines  issuing  from  tin/  oii^-in  would  the  ^ivcu  inte-ral  ciiuvcrij-e '.' 


ON  INFINITE  INTEGRALS  367 

^ —  =  ^^ —  To  integrate  about  z  =  —  1  use  the  binomial 

0    (1  +  X)-         sin  air 

expansion   z'^-i  =  [-  1  +  1  +  2]«-i  =  (-  l)'^-i[l  +  (1  -  a)  (1  +  z)  +  ,,(1  +  z)], 
t;  small. 

4.  Integrate  e-^"  around  a  circular  sector  with  vertex  at  z  =  0  and  bounded  by 
the  real  axis  and  a  line  iiulincd  to  it  at  an  angle  of  \tt.    Hence  show 


2 

1       \ir 


e*      I      (cos  r- —  i  sin  r-)  cZ>- =  |      c-^'dx, 

J^  50  ^  CO  1  /tT 

cosx^fZx  —  I     sinx-dx  =  -  \\~. 
0  Jo  2X2 

5.  Integrate  e-^''  around  a  rectangle  ?/  =  0,  y  =  5,  x  =  J-  .1,  and  show 

I     e-  '■'-"'  cos  2  axdx  =  I  Vttc-  «',  /     e-  -''^  sin  2  rtXcZx  =:  0. 

6.  Integrate  z'^-^e-^,  0  <  a,  along  a  sector  of  angle  q  <Iit  io  show 

seca^  I     x'^  - '^e-^'^'^^1  cos  {x  sin  q)dx 

—  CSC  aq  I     x"~'^e-^''''^ism  {xsmq)dx=  (     x"-'^e-^t 
Jo  Jo 


dx. 


7.  Establish  the  following  results  by  the  proper  change  of  variable  : 

,     r=°cosa:x   ,        7re-«*  „  ,„     f'"x'^-V/x      7r/3«-i     „^  ^ 

a)    I      dx  =  ,   a>0,  {13)  =  ~ '  /3  >  0, 

Jo    X-  +  k"  2  k  Jo       ji  +  X        sm  trTr 

J^'",.,  1/—  ,/""  1,  /''■ 

0  2  a  J'  Vx  ^^^ 

—     J^ 

J^^       „  ,  vTre    ■4"^  /•  1       dx  / — 

e-  «'■'■'  cos  /ixdx  =  ,  a  >  0.  (f )    I =  Vtt, 
0                                         2(T                                     Jo    V-  logx 

^     /"=  cosx  ,         /'^"sinx  ,  'tt  ,,,      /^Mogxdx  tt 

V)    /      — ^dx=/      -— -dx  =  ^-,  (^)    /        H =  -- 

Jo      Vx  '^o      Vx  >  -  ^'J   V 1  -  x2  ^ 


--log  2. 


8.  By  integration  by  parts  or  other  devices  show  the  following : 

X  log  sin  xdx  = tt-  loir  2,  (fi)    i      '- dx  =  —, 

0  2  '  Jo        x'^  2 

f=°sinxcosax  tt  .,       ^  .         tt  ..  ,    ^         n  •*  i     i  ^  i 

7)    I      dx  =  -  if  —  1  <  a  <  1,  or  -  if  or  =  ±  1   or  0  if    q-  >  1, 

Jo  X  2  4 

f"°                00                A   TT                                                                      r "               .>  ^               3   V  TT 
x2e-«"-'~dx  = ,  (e)    I     x*e-'^--^"dx  = -, 

4  a3  ^   Mo  8  a- 

J^  30                                                                Z'  '''     T  sill  Xflj*  TT" 

x«-ie-^dx,         (r?)    I      -^^ ^  =  — 
0                                      Jo     1  +  cos'-  X       4 

r^       /         1\     dx 
0)    I     loic  ( X  +  - ) =  TT  log  2,  by  virtue  of  x  =  tan  y. 

Jo       '  \        x/  1  +  X- 


368  INTEGRAL  CALCULUS 

fix 


J"  ''-  fix 

f{x)  —  ,  where  a  >  0,  converges.    Then  if  p  >  0  ,  7  >  0, 
ti  X 

Show  nf(P^r)-fi,x)  ^^  ^  j.^^^    r'^'y^^^  >lx  ^j,^^^^  ,^^^.  ,  _ 

Jo  X  a  =  0  «/y<«  X  p 

Hence     (a)    I —  dx  =  0,  (/3)    / (Zx  =  log  -  , 

Jo  X  J')  X  p 

XI  .r''^i  —  .r'/-!  ,         ,       a                    r^  cos.r  —  cos  ".r  , 
cZx  =  log-i,        (0)    I       ^Zj:^l<.ga. 
luLi^r                           n                   Jm                  .f 


.1  j-;<-i  —  .r'/^i  ,         ,       (/  , .,     C^'  cos.r  —  cos'/.r 

10.   \i  f  {s)  anil/'(.f)  are  continuons,  sliow  by  intearation  by  parts  tlial 

^in  h\. 


liin     I     /■(,/•)  sin /i-.fdr  =  0.     Hence  prove      iini     |     ^'(./'l '~dx  —  ~f(()). 

k=  A    Jn  /■  =  -/-.    J'l  X  2 

\\  rite  f{x) dx  —  /^(O)  / f/j  +  | sinA-.fdj. 

!  J)  X  J<)         X  Jo  X  J 

Apply  Kx.  0,  p.  ooO,  to  prove  these  fornuilas  under  general  hypotheses. 
IL  Show  that  lini    f  f{x) "   ^  "'  dx  =  0  if  6  >  a  >  0.    Hence  note  that 

Xh          sin  A*c                               p^          sin  At 
f{x)' '-d:^]\m   lini    |     f(x)' dx,     unless    /(O)  =  0. 

144.  Functions  defined  by  infinite  integrals.  If  the  iiitogiand  of  an 
integral  eontains  a  |)ai'anieter  ( JJ  118),  tlie  integral  defines  a  function  of 
the  parameter  for  every  value  of  the  parameter  for  "whieh  it  eonvei'ges. 
The  continuity  and  tlie  differentiaVtility  and  integrahility  of  the-  func- 
tion have  to  l)e  treated.    Consider  first  the  case  of  a:!  infinite  limit 


/       ^(./•,  ii)<lr  =   j    f(.r,  n),/.r  +  /.•(,-•.  n),  11  =    I 


f(  ./•    n  )  <l.r. 


If  this  integral  is  to  converge  for  a  given  value  a  =  n\^,  it  is  necessary  that 
the  rt'Uiainder  /'  (./'.  'r  )  c-an  he  made  as  small  as  desirt-d  ])y  taking  ,/•  large 
enough,  and  sliall  I'cmain  so  for  all  larger  values  of  ,/•.  In  like  manner  if 
the  inteu^rand  Ix'comes  infinite  lor  the  \alue  .r  =  h,  the  condition  that 


/    /'(.'■,  't)>/.r  =    I     /'(.'•.  n),/.r  +  A' ( ./•.  a),  Ji  =    I    f(.r. 


") 


f/.r 


(■onv«'!-ge  is  that  /,'(./■.  '(  1  can  he  made  as  small  as  desired  liy  taking./' 
near  enough  to  A.  and  shall  r.uuain  so  for  nearer  values. 

Now  for  dirt'ei'ent  valiirs  of  n.  the  least  values  of  ,/■  which  will  make 
/•'(.'",  '( )    =5  e.  when  e  is  assigmd.  will  ]ii-()l)al)ly  differ.    The  infinite'  inte- 
grals are  .said  to  cofin'r^/r  u iiifuniili/  for  a  range  of  values  of  a  such  as 


ON  INFINITE  INTEGRALS  369 

n^^  =  a  =  'i^  -when  it  is  }»ossil)le  to  take  x  so  large  (or  x  so  near  h)  that 
\  It  (x,  «)  j  <  e  liolds  (and  continues  to  hold  for  all  larger  values,  or  values 
nearer  I/)  simultaneously  tor  all  values  of  a  in  the  range  a^  =  «  =  a^. 
The  most  useful  test  foi'  uniform  convergence  is  contained  in  the 
theorem:   //' "  jnniiflri'  function  <^(.'')  can  hi'  fonnd  such  that 


f 


(j)  (./•)  dx     converges  and     eft  (x)  ^  \f(^x,  a)  | 


fovall  bii'fjc  ralues  of  x  and  for  all  nilnt's  <f<i  ni  thu  interval  'f^  =  <i  =  'f,, 
the  integral  af  f{j-,  a)  to  inp'nif;/  conrcrgcs  an'ifornihj  {<ind  ahsohitcli/) 
for  the  range  of  values  in  a.    The  proof  is  contained  in  the  relatid;; 


r  f{x,a)dx\^    C  4>(x)dx< 


■which  holds  for  all  values  of  a  in  tlie  range.    There  is  clearly  a  similar 
theorem  for  the  case  of  an  infinite  integrand.  See  also  Ex.  18  below. 

Fundamental  theorems  are :  *  Over  any  interval  a^^  a  ^  a  wlu^-e 
an  infinite  integral  converges  uniforndy  the  integral  defines  a  con- 
tinuous function  of  a.  Tins  function  may  l)e  integrated  over  any  finite 
interval  where  the  convergence  is  uniform  l)y  integrating  with  respect 
to  a  under  the  sign  of  integration  Avith  I'espect  to  :/•.  The  function  may 
be  differentiated  at  any  })oint  <i^  of  the  interval  ''<„  =  'f  =  <t^  hy  differ- 
entiating Avith  respect  to  a  under  the  sign  of  integi'ation  Avitli  respei^t 
to  X  provided  the  integral  ol)tained  l)y  this  differentiation  converges 
uniformly  for  values  of  a  in  the  neighborhood  of  a^.  Proofs  of  these 
theorems  are  given  immediately  Ixdow.  t 

To  pr(jve  that  tlie  function  is  continuous  it'  the  convergence  i.s  iniiforni  let 

f{a)  =  I     /(.f,  cx)dx  =  I    /(/,  a)dx  +  A' (x,  a),         «„  =  a  s  a^, 

f{a  +  An-)  =   r  fix.  a  +  A(r)f/.r  +  iH-r,  a  +  Aa), 

J  a 

lA-/-!  ^     I     [/(/.  a  +  A<()  -/(./-,  cx)]dx    +  \  U{x,  a  +  Aa)  ]  -|-  ;  ll{x.  a)\. 

*  It  is  rif  course  assunifnl  tliat  /■('.'•,  a)  is  continuous  in  (.c,  a)  for  all  values  of  x  anrl  a 
under  consideration,  and  in  the  tlieorem  on  differentiation  it  is  further  assumed  that 
/^j  (.J-,  a)  is  continuous. 

t  It  should  he  noticed,  however,  that  although  the  conditions  which  have  heen 
imposed  are  xnijirlp/if  to  estalilish  the  theorems,  they  are  not  n ece ■•<■'<(.( r >/ ;  that  is,  it  may 
happen  that  the  function  will  he  continuous  and  that  its  derivative  and  integral  may  he 
obtained  hy  operating  under  the  sign  although  the  convergence  is  uf)t  uniform.  In  this 
case  a  special  investiLiatifin  wf)nld  have  to  he  undertaken  :  and  if  nfi  process  for  justifying 
the  continuity,  integration,  or  differentiation  could  he  devised,  it  miuht  he  necessary  in 
the  case  of  an  intet^ral  occurring  in  some  application  to  assume  that  the  formal  work  le(l 
to  the  i'ii;ht  result  if  the  result  looked  reasf)nah!e  from  the  point  fif  view  of  the  problem 
under  discussion,  — the  chance  of  getting  an  erroneous  result  would  he  tolerably  small. 


370  INTEGRAL   CALCULUS 

Now  let  X  be  taken  so  large  that  |fi|<e  for  all  a"s  and  for  all  larger  values  of  x 
—  the  condition  of  uniformity.    Then  the  finite  integral  (§  118) 

f{x,  a)dx    is  continuous  in  a  and  hence        /     [/(■^»  «  +  Aa)  — /(j,  a)]dx 

a  \Ja' 

can  be  made  less  than  e  by  taking  Aa  small  enough.  Hence  j  Af  |<3e;  that  i.s,  by 
taking  Aa  small  enough  the  quantity  |  Af  |  may  be  made  less  than  any  assigned 
number  3e.    The  continuity  is  therefore  proved. 

To  prove  the  integrability  under  the  sign  a  like  use  is  made  of  the  condition  of 
uniformity  and  of  the  earlier  proof  for  a  finite  integral  (§  120). 

C"'xp{a)da  =  f'  r /(x,  a)dxda  +  f'Udx  =  f    ["'/{x,  a)dadx  +  f. 

Now  let  X  become  infinite.  The  quantity  f  can  approach  no  other  limit  than  0; 
for  by  taking  x  large  enough  E  <  e  and  |f  |  <  e  {a^  —  a^,)  independently  of  a.  Hence 
as  X  becomes  infinite,  the  integral  converges  to  the  constant  expression  on  the 

left  and  _  n^  C''^ 

I     f  (a)  da  =  I       I     /(x,  a)  dadx. 

Ja^  Ja      Ja^ 

Moreover  if  the  integration  be  to  a  variable  limit  for  a.  then 

^{a)=  f  yp{a)da=  f     f  f{x,  a)dadx  =  f    F{x,  a)dx. 

F(x,  a)dx'  =  \   \       I    /{x,  a) dadx \=  •   \       I     /(•'',  «) dxda  <  e (a  —  a^). 


Al 


Hence  it  appears  that  the  remainder  for  the  new  integral  is  less  than  e  (a,  —  ag) 
for  all  values  of  a;  the  convergence  is  therefore  uniform  and  a  second  integration 
may  be  performed  if  desired.  Thus  if  an  infinite  integral  converges  uniformly,  it  may 
be  integrated  as  many  times  as  desired  under  the  sign.  It  should  be  noticed  that  the 
proof  fails  to  cover  the  case  of  integration  to  an  infinite  upper  limit  for  a. 
Tor  the  case  of  differentiation  it  is  iiecessary  to  show  that 

I     f  {x,  a^)dx  =  (p' (a^).  Consider  |     f'^{x.  a)dx  —  ui{a). 

As  the  infinite  integral  is  assumed  to  converge  uniformly  l)y  the  statement  of  the 
theorem,  it  is  x'ossible  to  integrate  with  respect  to  a  under  the  sign.    Then 

C  "w  (a)  da  =    f^  f"f'^  (x,  a)  dadx  =    T'  [/(x,  a)  -  fix.  a^)]  dx  =  <p  {a)  -  4>  (a^). 

Jay  Ja      Jay  Ja 

The  integral  on  the  left  may  be  differentiated  with  resi>ect  to  a.  and  hence 
<p{a)  nuist  be  differentiable.  The  differentiation  gives  oj(a)  =  <p'{a)  and  hence 
w(at)  =  (p'  (at).  The  theorem  is  therefore  proved.  This  theorem  and  the  two 
above  could  be  proved  in  analogous  ways  in  the  case  <>f  an  infinite  integral  due 
to  the  fact  that  the  integrand  /(x,  a)  became  infinite  at  the  ends  of  (or  within) 
the  interval  of  integration  with  respect  to  x;  the  prrxifs  need  not  be  given  here. 

145.  Tl)e  inetliod  of  integrating  or  differentiating  imder  tlie  sign  of 
integration  may  be  a})plied  to  evalnate  infinite  integrals  wlien  the  condi- 
tions of  uniformity  are  properly  satisfied,  in  precisely  the  same  jnainier  as 
the  inethod  was  previously  applied  to  the  case  of  finite  integrals  where 


ox  INFINITE  INTEGRALS  3T1 

the  question  of  the  uniformity  of  convergence  did  not  arise  (§§  119-120). 
Tlie  examples  given  below  will  serve  to  illustrate  how  the  method  works 
and  in  })articular  to  show  how  readily  the  test  for  uniformit}'  may  be 
applied  in  some  cases.  Some  of  the  examples  are  purposely  chosen  iden- 
tical Avith  some  whicli  have  previously  been  treated  by  other  methods. 

Consider  first  an  integral  which  may  be  found  by  direct  integration,  namely, 

Jr>  r.                                                          d  p-r.  J 

e-  «-^  cos  hxdx  — Compare    (     e-  '^dx  =  -  . 
0                                a^  +  h"  Jo  a 

The  integrand  e-'^-^  is  a  jjositive  quantity  greater  than  or  equal  to  e-'^^coahx 
for  all  values  of  b.  Hence,  by  the  general  test,  the  first  integral  regarded  as  a 
function  of  h  converges  uniformly  for  all  values  of  h.  defines  a  continuous  func- 
tion, and  may  be  integrated  Ijctween  any  limits,  say  from  0  to  b.    Then 

i      I      e-"-'co-ibxdj:db  =  I       |     e-^^coabxdbdx 

J  0    J  0  J  0      J  0 

r  '^-  sin6x  ,  /•''     ndJ)  ,  b 

—  I      e-"-'^ ax  —  I =  tan-i-. 

Jo  X  Jo   «-  +  b'^  a 

sin/>x  „  ,         r'^'          1  — cos/>x  , 
■'■ ax 


P  '^'   P'>  sm  ox  c 

Integrate  again.     |        I    (•-'"■  ^ dbdx  =  j 

J  0     J  0  X  Jo 


b       a 

=  b  tan-i log  («-  +  />'). 

a      2 

1  —  cos  5x  ,  ,       /•  ^  1  —  cos  bx.  , 

dx. 


r*      „  .  1  —  cos5x  r^  1 

Compare  |      e-  "■' ax     and      |      - 

Jo  x~  Jo 

Now  as  the  sec(jnd  integral  has  a  positive  integrand  which  is  never  less  than  the  inte- 
grand of  the  first  fur  any  positive  value  of  a,  the  first  integral  converges  uniformly 
for  all  positive  values  i)f  a  including  0,  is  a  continuous  function  of  a,  and  the  value 
of  the  integral  f(U-  a  =  0  may  be  found  by  setting  a  equal  to  0  in  the  integrand.  Tlien 


r '  1  —  CHS  bx ,     ,.   r,      ,  b    « ,    ,  ,    , ,  n 

I      dx  =  lim    b  tan-i log  (a^  +  b-) 

Jo  X-  a  =  oL  a      2     '  J 


'  2 


The  change  of  the  varialjle  to  x'  =  1  x  and  an  integration  by  ijarts  give  respectively 

J"'^- s\n-bx  ,         TT ,-,  .             r'^'-iiinbx  ,              tt                  tt  ,      ^  ,      ^ 
dx  =-\b\          I      dx=  +  ~     or .     as     '^  >  0     or     b<0. 
0        X-               2  '    '           Jo         X                     2                  2 ' 

This  last  result  might  be  obtained  formally  by  taking  the  limit 

r  ^-           sin />x  ,         r  "^  sin /jx  ,  Ij  tt 

lim    (      e-nc ax  =  |      dx  =  tan-^-=  ±  — 

a  =  ',Jo  X  Jo  X  0  2 

after  the  first  integration  ;  but  such  a  process  would  be  unjustifiable  without  first 
showing  tliat  the  integral  was  a  continuous  function  of  a  for  small  positive  values  of  a 
and  for  0.  In  this  case  ^ x  "^  e ~  "■''  sin  5x  ]  =  | x  "^  sin  x  |,  but  as  the  integral  of  | x~i  sin  bx  \ 
does  not  converge,  the  test  f(jr  uinformity  fails  to  apply.  Hence  the  limit  would  not 
be  justified  without  special  investigation.  Here  the  limit  does  give  the  right  result, 
but  a  simple  case  wliere  the  integral  of  the  Hunt  is  not  the  limit  of  the  integral  is 

,.        r'sin/>x  ,         ,.      /      7r\            TT        r",.     sin  te  ,      C  ^' ^^  ■,         r. 
lim    I      «x  =  Inn  (  ±  ^  I  =  ±  —  ^  /      Inn dx    I      -dx  =  0. 

ftioJo  X  6  =  0  \       2/  2        Jo      h  =  0       X  Jo      X 


372  l^TEGKxVL  CALCULUS 

As  a  second  example  consider  the  evaluation  of   I     e~\^^)dx.  Differentiate. 

da  Jt)  Jo  \        xj  X 

To  justify  the  differentiation  this  last  integral  must  be  siiuwn  to  converge  uni- 
forndy.  In  the  first  place  note  that  the  integrand  does  not  become  infinite  at  tlie 
origin,  although  one  of  its  factors  does.  Hence  the  integral  is  intinite  oidy  by  vir- 
tue of  its  infinite  limit.    Suppose  a  s  0  ;  then  for  large  values  of  x 

e    ^      ^/    M ]  ^  e -'"(-■'■'     and      |      e-^'dx     converges  (§  143). 

Hence  the  convergence  is  uniform  when  a  s  0.  and  the  differentiation  is  justified. 
But,  by  the  change  of  variable  x'  =  —  a/x,  when  a  >  0, 

Jo  X-       Jo  Jo 

Hence  the  derivative  above  found  is  zero  ;  (f>' {<<)  =  0  ami 

(p((i)  —  i     e    ^      ■'■'  dx  =  const.  =  I     c--''dx  =  },  Vtt  ; 
Jo  Jo 

for  the  integral  converges  uniformly  when  «  s  0  and  its  constant  value  may  be 
obtained  by  setting  a  =  0.  As  the  ccjiivergeuce  is  uniform  for  any  range  of  values 
of  a,  tlie  function  is  everywhere  continuous  and  equal  to  l  ^  ir. 

As  a  third  example  calculate  the  integral  (p(h)  =  I     (■-""-''' cos //xJx.    Xow 

Jo 


J"  *               ,  »                         1    r       „  „            1  '^         h       r  -^.y  „ 
—  xt'-''"-''siu  hfdx  = ,    (;'"'''  sin  Iix       ■ |      e~'^'''  cos  i 
0                                           "2  a-  L                       Jo       2  a-  ^^o 


The  second  stej)  is  obtained  by  integration  by  })arts.  The  previous  differentiation 
is  justified  by  the  fact  that  the  integral  of  .ct-"'''.  which  is  greater  than  the  inte- 
grand of  the  derived  integral,  converges.    'I'he  differential  ecjuation  may  be  solved. 

d4>  _         h  '■' 

dh 


(/),  (p  --  Cc    ia\  0(0)=   (      (-"■■'-dx=  — 

'2  a'-  J  I)  2  < 


Hence  0  (Jj)  =  <p  (0)  r~  4«-  =   f 

Jo 


IS  1,.1-d.i 


2  a 


In  deternnning  the   constant  ('.  the   funi'lion  (p{ii)   is  assumed  continuous,  as  the 
integral  for  (p  {!/)  obviously  eonvurges  uniformly  for  all  values  of  h. 

146.    The   (]U('sti()ii   of  the  intcL;'ratioii    iuhIci'  Iho  sis^'ii   is   iiatiinilly 
connected  with  the  (|tu'stiuii   of  intinite   (loul)l('  intc.yrals.    Tlic  doiihlc 


intfo'i-al     I  ./'(•''.  //)"'.l    o\'er  an  area  A   is  said  to  lie  an  intinite  inti'i^-ral 

if  tliat  area,  extends  out  iiidt'tinitcly  in  any  dii'cetion  or  if  tin'  function 
./'(•''?  ,'J)   l>eeonn'S   iidinitc  at  any  jioint  (jf  tin-  area.     The  dtdinition  of 


ox  INFINITE  INTEGRALS  373 

convergence  is  analogous  to  that  given  before  in  tlie  case  of  infinite 
simple  integrals.  If  the  area  -1  is  infinite,  it  is  replaced  by  a  finite 
area  A'  which  is  allowed  to  expand  so  as  to  cover  more  and  more  of 
the  area  .4.  If  the  function /(.r,  ?/)  becomes  infinite  at  a  point  or  along 
a  line  in  the  area  A,  the  area  .1  is  replaced  by  an  area  A'  from  which  the 
singularities  of  /(;/•,  //)  are  excluded,  and  again  the  area  .4 '  is  allowed  to 
ex})and  and  approach  coincidence  with  A.  If  then  the  double  integral 
extended  over  A'  approaches  a  definite  limit  Avhich  is  independent  of 
how  A'  approaches  A,  the  double  integral  is  said  to  converge.    As 


//^(•"■">"-"-"=//k('^-) 


/(</>,  ij/)  dudv, 


where  x  =  <f>(u,  r),  ?/  =  >/'(",  i'),  is  the  rule  for  the  change  of  varialjle 
and  is  applicable  to  .1',  it  is  clear  that  if  either  side  of  tlie  ecpiality 
approaches  a  limit  which  is  independent  of  how  A'  approaches  A,  the 
other  side  must  approach  the  same  limit. 

The  theory  of  infinite  double  integrals  presents  numerous  difiiculties, 
the  solution  of  which  is  beyond  the  scope  of  this  work.  It  will  be  suffi- 
cient to  point  out  in  a  simple  case  the  questions  that  arise,  and  then 
state  without  proof  a  theorem  which  covers  the  cases  which  arise  in 
practice.  Suppose  the  region  of  integration  is  a  complete  quadrant  so 
that  the  limits  for  x  and  //  are  0  and  cc.  The  first  question  is,  If  the 
double  integral  converges,  may  it  be  evaluated  by  successive  integra- 
tion as 

ff(x,  y)dA  =  f        f    fix,  y)dydx  =  f        f    f(x,  >/)dxdi/? 

And  conversely,  if  one  of  the  iterated  integrals  converges  so  that  it  may 
be  evaluated,  does  the  other  one,  and  does  the  double  integral,  converge 
to  the  same  value  ?  A  part  of  this  question  also  arises  in  the  case  of  a 
function  defined  by  an  infinite  integral.    For  let 

'^(■'■)=  /      /(:<-,  y)^^y     and     /      cf>(x)dx=l         j      f{x,y)dydx, 

it  being  assumed  that  (^(x)  converges  exce])t  possibly  for  certain  values 
of  X,  and  that  the  integral  of  ^(.r)  from  0  to  :o  converges.  The  question 
arises,  ]\Iay  the  integral  of  <^(.f)  be  evaluated  by  integration  under  the 
sign  ?  The  proofs  given  in  §  144  for  uniformly  convergent  integrals  inte- 
grated over  a  finite  region  do  not  apply  to  this  case  of  an  infinite  inte- 
gral. In  any  particular  given  integral  special  methods  may  ]iossil)ly  be 
devised  to  justify  fur  that  case  the  desired  transfonnations.  lUit  niost 
cases   are   covered  by  a  theorem   due  to  de  la  ^'alle'   Poussin  :   If  the 


374  INTEGRAL  CALCULUS 

ftciiction  f(.r,  ij)  docs  not  cliangc  sl'jn  and  is  ronfhii/oHs  e:rc,ept  over  a  finite 
numhei'  of  lines  parallel  to  ilie  axes  of  x  and  //,  tlien  the  three  integrals 


ff{x,y)dA,        f        f\f(cr,y)d>/dx,        f        f  \f(.r,  y)dxd>/,     (12) 

cannot  lead  to  different  determinate  results  :  tliat  is,  if  an//  tiro  of  them 
lead  to  definite  results,  those  results  ((re  equ((l.*  The  chief  use  of  the 
theorem  is  to  estuhlish  the  equality  of  the  two  iterated  integrals  when 
each  is  known  to  converge ;  the  application  requires  no  test  for  uni- 
formity and  is  very  sim})le. 

As  an  example  of  the  use  of  the  theorem  consider  tlie  evahuition  of 

1=1     e-''J.c  =  I      ae-^'-'-'  tie. 
Jo  Jo 

Multiply  by  e-"'  and  integrate  from  0  to  x>  with  respect  to  a. 

Ze-«'=  r    ae- '^-(1 +•'■■')  tZx,       if    c-'^- da  =  1~  =  \       f    ac-'^-('^''-'-)  dxda. 

Ji)  Jo  Jo        Ji) 

Now  the  integrand  of  the  iterated  integral  is  positive  and  the  integral,  being  equal 
to  1'-,  lias  a  definite  value.    If  the  order  of  integrations  is  changed,  tlie  integral 

I        I      ixe-"-'<^^'-'~>  dadx=  \ =  -tan-ix  =  -- 

Jo     Jo  Jo     1  +  X-   2        2  4 

is  seen  also  to  lead  to  a  definite  value.    Hence  the  values  Z-  and  ]  tt  are  equal. 

EXERCISES 

\.  Xote  that  the  two  integrands  are  continuous  functions  of  (.r.  a)  in  the  whole 
region  0  ^  ^^  <  x,  0  ^  x  <  x  and  that  for  each  value  of  a  the  integrals  converge. 
Establish  the  forms  given  to  the  remainders  and  from  them  show  that  it  is  not  i)os- 
sible  to  take  x  .so  large  that  for  all  values  of  a  the  relation  |  R  (x.  a)  |  <  e  is  sati-sfied, 
l)Ut  may  be  .satisfied  for  all  a"s  such  that  0  <  (t,j  ==  a.  Hence  infer  that  the  conver- 
gence is  nonuniform  about  a  =  0.  but  uniform  elsewliere.  Note  that  the  functions 
defined  are  wot  continuous  at  a  =  0.  but  are  contimious  for  all  other  values. 

e-"'  -  1. 


Jo  J  a 

,   ^     /"  sin  tr.r  ,       -,,  ,        r  '  Aw  ax  ,  /•  ^  sin  . 

(/3)     / dx.ll{x.a)=\  dx=\ 

Jo  X  J -!■  X  J  a.r        X 


dx. 


2.  Kepeat  in  detail  the  i^roofs  relative  to  contiiuiity.  integration,  and  differ- 
entiation in  case  the  integral  is  infinite  owing  to  an  infinite  iiUegrand  at  x  =  h. 

*  The  theorem  may  hi-  gciicralizcd  by  allowing  /'(.r,  y)  to  be  discontimnms  over  a 
finite  mind)er  of  curvi's  rach  <>f  which  is  cut  in  only  a  finite  limited  number  of  ))uints 
l)y  lines  i)arallel  to  tlie  axis,  ^run-over,  the  function  may  clearly  he  allowed  ti)  cli.-uige 
sign  to  a  certain  extent,  as  in  tlii-  c;ise  where  /'>n  when  ./•  >  n.  and  /' <  0  -when  (^>  <  .'•  <  *', 
etc..  w  hei-e  the  integral  oNcr  the  wlmle  region  iiia>-  he  resdlved  into  the  sum  of  a  tinite 
number  of  integrals.  Finall.\-.  if  tlie  integrals  are  alisdlutely  cinnergeiit  auil  the  integrals 
of  !/'(.'•,  //)'  lead  to  definite  results,  so  will  the  integrals  ofy'(.;-.  //). 


ox  INFINITE  INTEGRALS  375 

3.  Show  tliat  differentiation  under  tlie  sign  is  allowable  in  tlie  following  cases, 
and  hence  derive  the  results  that  are  given  : 

.     r"^-         ■>  ,        1&  r.      r"^-    .  -  ,        ^^7^  1  •  3  •  •  •  (2  n  -  1) 

^0  ^  \'  a  '    ^0  ^  -"«"  +  ^ 

J^  •^-            „             1                       /•  *                      ,           1  ■  2  ■  ■  •  n 
xe-''^dx  =  — ,   or  >  0,     (     X'''+^  e-^^'-dx  = , 
u                           '2  a                   Jo                                      2a''"!"i 

^     r"^      dx          TT    1        ,       ^       r^          dx              tt  1  •  3- •  •  (2  h  —  1) 
7) = = ,  A;  >  0,     I       = , — '- , 

Jo     x^  +  k      2  v'^  J'J     (x-  +  i-)«+i       2        2"n:/j«  +  2 

x"(Zx  = ,  ?i  >  —  1.     I    x"(—  loK  xydx  = , 

0  n  +  l  Jo  '  (n+  l)"'+i 

e  )     I       dx  = 0  <  a:  <  1.    I       '—  dx  = 

Jo      1  +  X  sin  air  .^u  1  +  x  cos-  air  —  1 

4.  Establish  the  right  to  integrate  and  hence  evaluate  these  : 

p  X  p  X   ^—  ax  —  g—  bx  fj 

a)     (      e-"^dx,    0  <  a,,  =  a.    /       dx  =  los;  -,  h.  a  s  a„, 

Jo  Jo  X  "a 


/» 1  p  \  j'ft  ^  6 

j3)     I    x'^dx.    —  1  <  a,)  <  cr,    ( cZx  =  lO; 

Jo  Jo       log  X 


a  +  1     ,        ^ 

-,   />.   (/  ^  or,,. 


'    ^^  +  1 


^  X  /^  X    p—  CLT  p—  bx 

y)     I      e- «^' cos  ??jxdx,   0  <  (Xq  ^  tr.    |       cos?nxdx 

«/o  Jo  X  2     '  a-  4-  »(- 

/>  X  />  X   g—  ftr g—  bx  ]f  f-f 

5)     I     e-<^^sin7nxcZx.   0<cro^a.    |      sin  ??ixdx  =  tan-i tan-i   -, 

Jo  '  Jo  X  1)1)11 

e)     (     e-<^'^dx  =  — ~  ,  0  <  a„  ^  a.    f    e"j--  —  e    x^dx  =  (b  —  a)  Vtt. 
Jo  2  a  Jo 

5.  Evaluate:  (a)     f    e-'^'^^^^^^-^  dx  =tan-i -, 

Jo  X  a 

^      /^*        1  —  cos  n-x  ,        ,        /- ,        /^-^       oSin2arx, 

^)     I     e-^ dx  =  logVl  +  a-.         (7)     (      e-^' dx, 

Jo  X  Jo  X 

^,     f '-  -(x"-  +  '^)  ,         ^''^      ,.,        ^^  ^^     r-  log(l  +  «2x-^)  , 

5)     I      e    V       a-'(:Zx  = e--".    «  s  0.  (e)     (       — dx. 

Jo  2  Jo  1  4-  b-x'^ 


6.   If  0  <  a  <  b,  obtain  from  j      e-'"-''''dx  =  -  *   —  and  justify  the  relations : 
Jo  2   V  '■ 

X'' sin  ?•  2      f^  n  ^-         „  2      n^   r^        « 

— zr  dr  =  — -   I      /      e-  '■■'■'sin  rdxd)-  =  — =   I       I    e-  ™'sin  rdrdx 
..       ^   f  ^-.Ja    Jo  -^\jj.Jo      Ja 

-    r  •        r  "  e-  "^"x-dx        .    ,   r  ^  e-  ^'^'x-dx 

=  — 1;    sm  a sm  b  |      

Vtt  L  Jo        1  +  X*  Jo       1  +  x^ 

px(,-ax\jj.  f^-e-^'"dx~\ 

+  cos  a cos  5  I     . 

Jo      1  +  x*  Jo     l  +  x*J 

/""sin?-  ,  TT         2    r  .        />  ^  e- '■•'■'x-dx  r^- e-"'dx~] 

I      dr  =  \   sm  /■ 1-  cos  r  |       • 

Jo     ^r.  \2       ^"L         Jo        l+x*     ^  Jo      l  +  x*J 


376  INTEGRAL  CALCULUS 

,  .     .,     ,         /•'"cosr  ,  jir      2  Y  c^-  e-^'^-x^x        .        r- =^  g- '"■'^^dx'l 

Similarlv,     I     — =-  dr  =  -s^i cds  r         sin  r         . 

Jo     Vr  ^2       ^L         ^'^       l  +  x'^  Jo      l  +  x*J 

1  r'^ 

7.  Given  that =  2  (      ae- "^d  +  ^'Wa,  show  that 

1  +  x^         Jo 

J'^*  1  +  citamx  -,        7r ,,             ,           ,    /"'^  cosmx  ,        tt 
— dx  =  -  (1  +  e-  "')     and    |      dx  =  -  e- '",     m  >  0. 
0         1  +  x^               2  ^               '             Jo      1  +  x^  2 


X'-c  J*  sin  cxx 
-— ^  (ix,  by  integration  by  parts  and  also  by  substi- 
tuting x'  for  ax,  in  sucli  a  form  tliat  the  uniform  convergence  for  a  sucli  tliat 
0  <  a^  s  a  is  sliown.    Hence  from  Ex.  7  prove 

r^^'  xsin  ax  ,        tt  _  ,,      ,.^  .     .     ^ 

I      ^ax  =  — e-*^,         a>0  (by  differentiation). 

Show  that  this  integral  does  not  satisfy  the  test  for  uniformity  given  in  the  text; 
also  that  for  a  =  0  the  convergence  is  not  uniform  and  that  the  integral  is  also 
discontinuous. 

9.  If /(x,  a,  /3)  is  continuous  in  (x,  a,  ;3)  for  0  ^  x  <  x  and  for  all  i)oints  (a,  /3) 

of  a  region  in  the  aj3-plane,  and  if  the  integral  0  (a,  /3)  =    |     /(x,  a.  j3)dx  con- 
Jo       ■  ' 
verges  uniformly  for  said  values  of  (a.  /3),  show  that  <p  (a.  /3)  is  continuous  in  (a,  (3). 
Show  further  that  if /^(x,  a,  ^)  and/^(x,  a,  ^)  are  continuous  aiul  their  integrals 
c(jnverge  uniformly  for  said  values  of  (a,  p),  then 

j    /^(x,  a,  /3)(Zx  =  0^,  r  ./"^'Cx,  a,  /3)(Zx  =  0g, 

and  0^,  (p^  are  continuous  in  (a,  /3).    The  proof  in  the  text  holds  almost  verbatim. 

10.  If  /(x,  7)  = /(x,  a  +  i/S)  is  a  function  of  x  and  the  complex  variable 
y  =  a  +  (/3  which  is  continuous  in  (x,  a,  /3),  that  is,  in  (x,  7)  over  a  region  of  the 
7-plane,  etc.,  as  in  Ex.  9,  and  if /'(x,  7)  satisfies  the  same  conditions,  show  that 

0(7)  =r  I      f(x,  ■y)dx  defines  an  analytic  function  of  7  in  said  region. 

11.  Show  that  I  e-y''dx,  7  =  a  -f  1(3.  (t  ^  a,,  >  0.  defines  an  analytic  func- 
tion of  7  over  tlie  whole;  7-plane  to  the  right  of  the  \ertical  d  —  a^|.    Hence  infer 

0 (7)  =  r 'e- y-'-'dx  =  ^^''  =  -J-  ^—  ,         as, ,„ > 0. 
Jo  2    V  7       2    \  a  +  i(i 


Prove 


1       /tt  a  +  Va-  -I-  //- 


J'  •^             o                .-,  ,            1          TT  a  -{-A 
(■-  a.r-  Q^,^  Bx-dx  =  --  A     
u                                      2    \2         a- 

r^  ...         ., ,         1        TT  —  (f  -1-  Va-  -f  /i- 

/      (-"'-sinax-Uc  =  -  X — ^-^ 

Jo  2    \  2  a-  +  ^33 


ON  IXFIXITE  INTEGRALS  377 

-  e-<^^"x  cos  /Sx^dx  of  Ex.  11  by  parts  with  xcos^x^dx  =  du 
to  show  that  the  convergence  is  uniform  at  a  =  0.    Hence  find    /      cos^x-cZx. 

^   +   CO  /^  +  »  /.yj-  ^    +   CO 

13.  From    |         cosx-dx  =  I       cos  (x  +  a)2dx  = -^/- =  I        sin  (x  +  a)2dx,  with 

n+x  ^  +  CO 

the  results  I  cos  x- sin  2  crxdx  =  /  sin  x^  sin  2  axdx  =  0  due  to  the  fact  that 
sin  X  is  an  odd  function,  establisli  tlie  relations 

(      cos  x- cos  2  axdx  = cos  ( a-),     (      sinx^  cos2frxdx  =  sin  ( a'^]. 

Jo  2  \4  /     Jo  2  \4  / 

14.  Calculate:         (cr)    (      e- "'^■"  cosh  6xdx,  (/3)    (     xc- ""-^  cos  5xdx, 

Jo  I'u 

and  (together)  (7)  J^'cos  (^~  ±  ^^  dx,         (5)  £  "sin  (^^  ±  ^^  dx. 

15.  In  continuation  of  Exs.  10-11,  p.  308,  prove  at  least  formally  the  relations: 

,.        f"   .,     sinA-x  TT  1    /^«         sinA:x 

lim    I     /(x) dx  =  -/(0),  Inn-   I     /(x) dx  =/(0), 

k=xJ-a  X  2  A  =  z7rJ-rt  X 

Jr>  k      n  <l                                                                        /^  <l        n  Jc                                                                     r»  <t                     Sill   A,'iC 
I     f  {x)  co^  kxdxdk  —  I       I    f  {x)  cos  kxdkdx  —  j     /(x)-^ dx, 

-   I      I     f  (x)  COS  kxdxdk  =  \\m  -   (     f{x)- ^  dx  =/(0), 

TT  Jo      J  -a  /(■=Qc7rJ~<e  X 

-   r  "^  r  ^  /(^)  COS  txdxdfc  =  /(O),        -   f   r  '  /(x)  cos  fc  (x  -  0  dxdk  =  /(<) . 

The  last  form  is  known  as  Fourier's  Integral  ;  it  represents  a  function  f{t)  as  a 
double  infinite  integral  containing  a  parameter.  Wherever  possible,  justify  the 
steps  after  placing  sufficient  restrictions  on/(x). 

/-■  X  I  ^  X  g—  ax g—  bx  I) 

16.  From  (      c--"J dy  =  -  prove  /      dx  =  log-  •    Prove  also 

Jii  x  Jo  x  a 

I     x"-^e-^dx    (     x'«-ie-^'dx 
Jo  Jo 

IT 

—  9     I        ,.-2n  +  2m-2g-r2(7j.2    j    "  sin '-" -l0  COS^"' -l0d0. 

Jo  Jo 

17.  Treat  the  integrals  (12)  by  polar  coordinates  and  show  that 

I  /{X',  y)  dA  =  I  ~  j     f[r  cos  0,  r  sin  0)  rdrd^) 

J  '  Jl)     Jo 

will  converge  if  |/|  <  ?—  --^-  as  r  becomes  infinite.  If /(x,  ?/)  becomes  infinite  at  the 
origin,  but  \f\<r-^  +  ^,  the  integral  converges  as  r  approaches  zero.  Generalizx' 
these  results  to  triple  integrals  and  polar  coordinates  in  space  ;  the  only  difference 
is  that  2  becomes  3. 

18.  As  in  Exs.  1,  8,  12,  uniformity  of  convergence  may  often  be  tested  directly, 
without  the  test  of  page  369  ;  treat  the  integrand  x-^e-^^sin  bx  of  page  371,  where 
that  test  failed. 


CHAPTER  XIV 

SPECIAL  FUNCTIONS  DEFINED  BY  INTEGRALS 

147.  The  Gamma  and  Beta  functions.    The  two  integrals 

r(»)=    r   ./■"-it'-V./-,  B(m,?i)=    I    x"'-\l -.>■)" -hlx  (1) 

converge  when  n  >  0  and  i/i  >  0,  and  lience  define  functions  of  the 
l)ai-anieters  n  or  n  and  m  for  ajl  positive  values,  zero  not  included. 
Other  forms  may  lie  obtained  by  changes  of  \ariable.    Thus 

T{7i)  =  2  fr"-'>'-'-n/>/,  by     x=>/%  (2) 

T(?i)=J  (log-J    \l>/,  by     e--  =  y,  (3) 

B(//^  72)  =f  y-Hl  -  y)"'-V/y  =  B(n,  m),      by     x  =  l-  >/,      (4) 

TJ/  X  f"         //"'-^^///  //  ... 

B(/M,  ?i)=   I      -::; :~  '  by     x  =  - ,      (o) 

B(m,7i)  —  2J   'sin'-'"-^c^cos-"-'<^f/0,  by     a- =  sin- (^.       (6) 

If  the  original  form  of  T(7i)  be  integrated  l)y  parts,  then 

X  '^  Jo        '^Jo  "' 

The  resulting  relation  V(n  +  1)  ==  nT(n)  shows  that  the  values  of  the 
F-function  for  n  +  1  may  be  obtained  from  those  for  n,  and  that  con- 
sequently the  vabu'S  of  the  function  will  all  be  determined  if  the  values 
over  a  unit  interval  are  known.    Furthermore 

r(  /;  +  1 )  =  nT(n)  =  n  (ii  -  l)T(n  -  1) 

=  nin  -  l)---(n-k)V(n-h)  ^  ^ 

is  found  by  successive  reduction,  where  /,■  is  any  integer  less  than  n. 
If  in  particular  n  is  an  integer  and  k  =  n  —  1,  then 

Via  +  1)-  n(n-  \)  ■  ■  ■  2  ■  1  -V  (V)  =  n\V  {1)  ^  n\ ;  (8) 

378 


FUXCTIOXS  DEFINED  BY  IXTEGFvALS  379 

since  wlien  n  =  1  a  direct  integration  shows  that  r(l)  =  1.  Thus /b?*  inte- 
gral ralurs  ofn  the  V-fuwtion  is  the  factorial ;  and  for  other  than  integral 
values  it  may  1)eT-egarded  as  a  sort  oT^eneralization  of  the  factorial. 

Both  the  F-  and  B-f unctions  are  continuous  for  all  values  of  the 
parameters  greater  than,  but  not  including,  zero.  To  prove  this  it  is 
sufficient  to  show  that  the  convergence  is  uniform.  Let  n  be  any  value 
in  the  interval  0  <  ??^  ^  ?i  s  ;\^;  then 

/  3.«-i^-T,/_,.  ^    I  ./"o-ie-^c/a:,  I     .r"-ie-V.r  ^   |     x^'-^e-'-'dx. 

The  two  integrals  converge  and  the  general  test  for  uniformity  (§  144) 
therefore  applies ;  the  application  at  the  lower  limit  is  not  necessary 
except  when  n  <  1.  Similar  tests  apply  to  B(//;,  n).  Integration  with 
respect  to  the  parameter  may  therefore  be  carried  under  the  sign.  The 
derivatives  d^viv^        C" 

-^=j^    x-'e-^(logxfdx  (9) 

may  also  be  had  by  differentiating  under  the  sign ;  for  these  derived 
integrals  may  likewise  be  shown  to  converge  uniformh". 

By  multiplying  two  F-functions  expressed  as  in  (2),  treating  the 
product  as  an  iterated  or  double  integral  extended  over  a  whole  quad- 
rant, and  evaluating  by  transformation  to  polar  coordinates  (all  of 
which  is  justifiable  by  §  14(),  since  the  integrands  are  positive  and 
the  processes  lead  to  a  determinate  result),  the  B-function  may  be 
expressed  in  terms  of  the  F-function. 

r(?7)r(??i)  =  4|    x-"-^e-^dx  I   7/2'»-ie-^'V//=4  /     /    x-"-hj-">-^e-'''-!''dxd>/ 

=  4  I     ,.2n  +  2m-ig-r2^7,.  /  ^sin^'''-^^  COS"" -^^r/<^  =  T (h  +  vi)B(m,  n). 
Hence  B  (,v,  n)  =  ^J'"^^^'')  =  B  (n,  w).  (10) 

The  result  is  symmetric  in  m  and  n,  as  must  be  the  case  inasmuch 
as  the  B-function  has  been  seen  by  (4)  to  be  symmetric. 

That  F a_)  =  Vtt  follows  from  (9)  of  §  143  after  setting  w  =  |  in  (2) : 
it  may  also  be  deduced  from  a  relation  of  importance  which  is  obtained 
from  (10)  and  (5),  and  from  (8)  of  §  142,  namely,  if  n  <  1, 

Tin)Va-n) 


=  B{n,l-n)=£^ylt 


r(l)=l 
or  r(n)F(l-n)=^    .  '^      ■  (i^j 


380  INTEGRAL  CALCULU8 

As  it  was  seen  that  all  values  of  r(?i)  could  be  had  from  those  in  a 
unit  interval,  say  from  0  to  1,  the  relation  (11)  shows  that  the  inter- 
val may  be  further  reduced  to  ^  ^n^l  and  that  the  values  for  the 
interval  0  <  1  —  Ji  <  ^  may  then  be  found. 

148.  IJy  suitable  changes  of  variable  a  great  many  integrals  may 
be  reduced  to  B-  and  F-integrals  and  thus  expressed  in  terms  of 
F-functions.  ]Many  of  these  types  are  given  in  the  exercises  below ; 
a  few  of  the  most  important  ones  will  be  taken  up  here.    By  ij  —  ox, 

j    x"'-'^('i,  —  .r/'-i^/,/:  =  (/"'  +  "-i  I     (/"'-\1  —  y'y^Oij  =  «"'  +  "-iB(7H,  n) 

or  r  x'"-Ha  -  ry-hlx  =  a'"  +  "-i  iX^liffl") ,        a  >  ^.  (12) 

Jo  Y{jn  +  n) 

Xext  let  it  be  required  to  evaluate  the  triple  integral 

/  =   fCCx'-hr-^z^-^by/udz,         ,r  +  i/  +  z^  1, 

over  the  volume  bounded  by  the  coordinate  planes  and  ./•  +  1/  +  '-'  =  Ij 
that  is,  over  all  positive  values  of  ./;,  y, ;:  such  that  x  +  y  +  '^  =  1-    Then 

nl  —  X        /^  1  —  X  —  1/ 
I  x'-'f'-'z"-'dzdi/dx 

=  -    /       I         x^-\/"'-\l  — X —  >/)"(/ /jdx. 


By  (12)  J"  V-^(l  -  ^-  -  1/)"'^!/  =  wI'lTT-tV  (1  -  •^O'"^"- 


V(m  +71  +  1) 


Then  /  =     \        ■         ,  '         ,/■'  "Ml-.'' )'"  +  "'f-r 

_  T(m)T(n  +  l}  T(l)T(ui  +  n  +  1) 
~  nT{ii>  +  n  +  1)  T(^l  +  m  +  71  +  1 )  ' 

This  result  may  be  simplified  by  (7)  and  by  cancellation.    Then 

JJJ  r(/  +  ,«  +  »+i)  ^ 

There  are  simple  modifications  and  _ireneralizations  of  these  results  whicli  are 
sometimes  nseful.  For  instance  if  it  were  desired  to  evaluate  /  over  the  ramxe 
of  positive  values  such  that  r/((  +  y/h  +  z/r  ^  A.  the  change  x  =  ah^.  y  —  bhr], 
z  =  ch^  gives 


C j  j  x'-^y"'-'^z"-hlxdy(lz  =  «'6'"c» 


r(/)r(m)r()0   ,,^„,,„      j-  ,  u 


+  T  + 


r  {l+m+  n+  I)  '       a      h 


FUNCTIONS  DEFINED  BY  IXTEGEALS  881 

The  value  of  this  Integral  extended  over  the  lamina  between  two  parallel  planes 
determined  by  the  values  h  and  h  -\-  dh  for  the  constant  h  would  be 

T(l)T(m)V{n)  ^.^     ^      ,„ 
dl  =  a'b"'c''  ^-^ — ^—^ — !^  h'  +  >"  +  «  -  ^dh. 
V  {I  +  m  +  n) 

Hence  if  the  integrand  contained  a  function /(/<),  the  reduction  would  be 
JJ/x'-i2/"'-i2«-i/  t^  +  l+^\  dxdydz 

T{1  +  m  +71)  Jo    ^  ' 

if  the  integration  be  extended  over  all  values  x/n  +  y/b  +  z/c  ^  //. 

Another  modification  is  to  the  case  of  the  integral  extended  over  a  volume 

j=fff"-'«-'-"='-'<'"'«<u,    @"+  0)'+  @'-  *, 

which  is  the  octant  of  the  surface  {x/a)p  +  {y/h)i  +  (z/c)''  =  Ji.    The  reduction  to 


?       m      1 

■  J :  - 


J  = 


a'b"'c"}u'    'I  ' '"  rrr  --i    -  -i   -  -i 


fff'''    ^  '^'^    ^  ^'    ^  '^^'-^V'J^^  I  +  'J  +  f  ^  1, 


pqr 
is  made  by   ^h  =  ('^Y,  Vi  =  (-J,  ^h  =  (~)  ,  dx  =  ~hl'^'i> 


J=  CfCx^-hy>"-iz"-hlxdydz 


r(-ir(~)r 


h  1'     'I     '•. 


(('b^'c"^      \p/      \(i/      \/7       --(-'"4-' 

-'  ^     r  -  +  -  +  -  +  1 
\p      q      r 


This  integral  is  of  impurtance  because  the  bounding  surface  here  occurring  is  of  a 
type  tolerably  fanuliar  and  frequently  arising  ;  it  includes  the  ellipsoid,  the  surface 

1111  ''222 

X2  4-  2/2  4-  z2  =  a2,  the  surface  X3  +  ys  +  zs  =  as.  By  taking  I  =  m  =  n  =  1  the 
volumes  of  tlie  octants  are  expressed  in  terms  of  the  T-function  ;  by  taking  first 
I  =  3,  m  =  71  =  1,  and  then  m  =  3,  Z  =  ?i  =  1,  and  adding  the  results,  the  moments 
of  inertia  about  the  z-axis  are  found. 

Although  the  case  of  a  triple  integral  has  been  treated,  the  results  for  a  double 
integral  or  a  quadruple  integral  or  integral  of  higher  multiplicity  are  made  oVtvious. 
For  example. 

xJ-Uf'-klxdy  = -i '—- hp    '1,  -     + 

J  J  P'l  r/l  +  !!i  +  i\  W 

\p        q  I 


r(-ir 


iJ----4©''- (;;)1-'- ^Tzf^  X'' ^'*>''^ '"■-• 


p     q 


r-(0'-"- 


;382  INTEGllAL  CALCULUS 


(") 

pqrs  (k       I 


r    -  +  -  +  -  +  ^  +  1 
\p      q       r       s        I 

12  n  —  1 

149.  If  the  product  (11)  be  formed  for  each  of  -  j  -  j  ■  •  ■•> ?  and 

n    n  n 

the  results  be  multiplied  and  reduced  by  Ex.  19  below,  then 

The  logarithms  may  be  taken  and  the  result  be  divided  by  n. 

1  \o<i  n 


l-KD-^G-^)-- 


Now  if  n  be  allowed  to  become  infinite,  the  sum  on  the  left  is  that 
formed  in  computing  an  integral  if  dx  =  1/n.    Hence 

Ihn  V  log  r  (,;•,)  A..;  =   f  log  V  {:r)  ,lr  =  log  V2^.  (15) 

Then  /    log  V  (<i  +  ,r)  dx  =  a  (log  a  —  1)  +  log  V2^  (lo') 

may  be  evaluated  by  differentiating  under  the  sign  (Ex.  12  (6),  p.  288). 
]>y  the  use  of  differentiation  and  integration  under  the  sign,  the 
ex])ressions  for  the  first  and  second  logarithmic  derivatives  of  r(«) 
and  for  log  r(w)  itself  may  l)e  found  as  definite  integrals.  By  (9) 
and  the  expression  of  Ex.  4  (<■(),  p.  375,  for  log  a-, 


T'(n)=  I     x"-h'-'\(v^xdx  =  I     x"-^e-''  I 


dadd 


If  the  iterated  integral  l)e  regarded  as  a  double  integral,  the  order  of 
the  integrations  may  be  inverted  :  for  the  integrand  maintains  a  ])osi- 
tive  sign  in  tlie  region  1  <  ,r  <  x,  0  <  a'  <  x,  and  a  negative  sign  in 
the  region  0  <  x  <  1,  0  <  cr  <  x,  and  the  integral  from  0  to  x  in  x 
may  be  considered  as  the  sum  of  the  integrals  from  0  to  1  and  from 
1  to  X,  —  to  each  of  Avhich  tlie  inversion  is  applicable  (§  146)  because 
the  integrand  does  not  change  sign  and  the  results  (to  be  obtained) 
are  definite.     Then  by  Ex.  l('t). 


T\n)  =   I        I      j:"-^-  '• dxda  =  T  {n)  j 


1       \  da 


(1  +  a)"/  a 


FUIS'CTIOXS  DEFINED  BY  IXTEGRALS 


383 


or 


This  value  may  be  simplified  by  subtracting  from  it  the  particular 
value  -  y  =  r'(l)/r(l)=  r'(l)  found  for  n  =  1.    Then 

r(n)  _  r(i)^  ^  r(n)  ,     ^  r  /_^ l      \  ^r 

r  {n)       r  (1)        r  («;  "^  ^     J^     \1  +  ^r       (1  +  «/'/   «  " 
The  change  of  1  +  a  to  l/«  or  to  e"^  gives 
I>) 

r(n) 


T'(n)  r '  1  -  ""  - '  ,  f  "  e-  '^  -  e- "" 


rfa. 


r/- 


Differentiate  :  -— -,  logr(«)=  | da. 

(/n-     °     ^  ''     J.      1  —  e-" 

To  find  log  r(«)  integrate  (16)  from  71  =  1  to  ?i  =  71.    Then 


(1") 
(18) 


iogr(»)  = 


(?i  —  l)e-"- 


log(l  +  a) 
since  r(l)  =  1  and  log  r(l)  =  0.    As  r(2)  =  1, 

a  +  nr' 


<hx 


^'      (19) 


iogr(2)  =  o  = 


da^ 


and   log  r  (?i) 


a         log(l  +  a)J      ' 
V  —  \  (1  +  ^i)-^  —  (1  +  a')" 


(1  + 


a 


d<. 


log  (1  +  (t) 


by  subtracting  from  (19)  the  (juantity  (n  —  1 )  log  V('2)  =  0.    Finally 

da 


logT(n)=J^ 


{u.-!)," 


(19') 


if  1  +  'r  be  changed  to  e~".    The  details  of  the  reductions  and  the  justi- 
fication of  the  differentiation  and  integration  will  l)e  left  as  exercises. 

An  approximate  expression  or,  better,  an  asi/mpfotir  expression, 
that  is,  an  expression  with  sumll  percentdr/e  error,  may  be  found  for 
T(n  -f-  1)  when  ?i  is  hn-ge.  Choose  the  form  (2)  and  note  that  the  inte- 
grand y-"^^'"""  rises  from  0  to  a  maximum  at  the  point  //■  =  n  +  \  and 
falls  away  again  to  0.  ^fake  the  change  of  variable  //  =  Va'  +  tr,  where 
rr  =  ?i,  +  \,  so  as  to  l)ring  the  origin  under  tlie  maximum.    Then 

r  (?i  -f-  1)  =  2  I         (  Va :  +  ?/■)•->-  "^ -  -  ^"^'"-""du; 

tV  -   -V  (I 

or  r(rt -f  1)  =  2a-^'>-''  e        v      Va^  d/r. 

Now         2  a  log  ( 1  -f  -~\  —  2  V^v/-  ^0,  —  V7i  <  ir  <  -r,. 


384  INTEGRAL  CALCULUS 

The  integrand  is  therefore  always  h'ss  tlian  «"'"',  except  when  w  =  0 
and  the  integrand  becomes  1.  Moreover,  as  tc  increases,  the  inte- 
grand falls  off  very  rapidly,  and  the  chief  part  of  the  value  of  the 
integral  may  l)e  obtained  by  integrating  l)etween  rather  narrow 
limits  for  ir,  say  from  —  3  to  +  3.  As  a  is  large  by  hypothesis, 
the  value  of  log(l  -\- ■ir/'y/<c)  may  be  obtained  for  small  values  of  vj 
from  IVIaclaurin's  Formula.     Then 


'(rt  +  l)  =  2«'^e--  r  ^---^a- 


>dw 


is  an  approximate  form  for  V(n-{-l),  wliere  the  quantity  e  is  about 
2  ir/^a  and  Avhere  the  limits  ±  <■  of  the  integral  are  small  relative  to  v  a. 
But  as  the  integrand  falls  off  so  rapidly,  there  will  \)e  little  eri-or  made 
in  extending  the  limits  to  c/d  after  dropping  e.    Hence  approximately 


r  (?i  -f  ]  )  =  2  a-"e-  "  I     e-  -"■'  (/tr 


or  r  (n  -j-l)=  V27r  («,  +  ^)"  + 1  e'  ("  +  i)(l  +  r,),       '  (20) 

where  rj  is  a  small  quantity  approaching  0  as  ?t  becomes  infinite. 

EXERCISES 

1.  Estal)li.sli  the  following  formulas  by  cliaiif^es  of  variable. 

(a)   T  (n)  -  a"    C    x"  -k-  '^^dx,    a  >  0,  (/3)    f  "  sin"  xdx  =     B  ( "  +     ,  - ) . 

(7)   B(n,  n)  =  :^i--"B(??,  J)  by  {(i),  (5)    C  x"'-'^(l  -  x-^y>-hlx=  lB{lm,  n), 

«)  f 

Jo        (x  +  «)"'  +  «  «"(1  +  r/)'"      ««{1  +  «)'»  r  (m  +  n)  x  +  a      1  +  a 


x'"-i(l -/)"-!  B(m.v)  ]  r(?n)r(u)   ^  ,        x  y 

ax  =  —    = jtake  — 

«"(1  +  r/)'"       a"{l  +  «)'"  I'  ("^  +  ») 

-1(1  -  x)"-!'?/  r(?n)r(n)        ,,  ?«/ 

—  take  X  =  — 


(I    [ax  +  b{\  —  x)]'"  -  "       (L">b"Y  {m  +  n)  a  (1  —  //)  +  ''// 

■  1 X"'  -1(1  -  x)"  -hlr  _    R  (m.  7i)  ^^^    f  1     x"dx      __  a  'tt  T  {h  n  +  I) 


r^x'"^'{\  —  x)"-'(tx        \i{m.n)  ,.^    f^     x"dx  at 

(77)      I =r ,  (u)      I        -  =  —   - 

Ji>         (6  +  cx)"'  +  "  h"{b  +  c)"^  J  (I    v'i_j-i        2 

(O    /    x'"(l-x")^r/x.=     B  ;,  +  !,. _._+^,     ^     I     ^.^^=,----_^  ^^    ^^. 

2.  From  T  (1)  ==  1  and  F  (^)  =  a  tt  make  a  table  of  the  values  for  every  integer 
and  half  integer  fi-oni  0  to  5  and  plot  the  eurve  //  =  T  (x)  from  them. 

3.  By  the  aid  of  (10)  and  Kx.  1  (7)  prove  the  relations 

\/7rr  (2  a)  =  22«-ir  (a)  T  (a  +  '),  V^V  {n)  =  2"-ir  (J  n)  T  (4  n  +  }.). 

4.  Given  that  T  (l.To)  =  O.Mim.  add  to  the  table  of  Kx.  2  the  values  of  T  (it)  for 
every  (jnarter  from  0  to  3  and  add  the  points  to  the  plot. 


FUXCTIOXS  DEFINED  BY  INTEGRALS  385 

5.  "With  the  aid  of  the  r-fimction  prove  these  relations  (see  Ex.  1)  : 

,    .     r\  .   „     .  r-2       „     ,         l-.3-o...(n-l)  TT  2.  4.  6...  (71 -1) 

{a)    I     s\\\"  xdx  =  I      cos"a;'Zx  = ^ '^"-  ' 

•Jo  Jo  2  •  4  •  0 


n        2  1  •  3  •  5  ■  •  •  71 

as  71  is  even  or  odd. 


.^,     ri    x-"d.r         1.3- o--- (2n  -  1)  TT         ,^     r  i  j'-^«+it?j            2-4-6...2u 
(P)  —  = .        (7)    I     — — 

'Jo^l^X^  2.4.0...:' 71  2  -'"      Vl-X^         1.3.  5...  (271+1 


) 

3  7r«6 


X-  V  «■-  —  X-dx  =  ,  (e)    j     X-{a-  —  x-)^dx  = 

0  10  Jo 

J^  1       r?x  p  ^ 

— ^;;=  to  four  decimals.  (7;)  Find    /     — 

0     -v/l  —  r*  Jo         / ,  ^ 

6.  Find  the  areas  of  the  quadrants  of  these  curves  : 

(a)  x-^  +  ?/2  =  a^,  (p)  xi  +  if:  =  en,         (7)  ,r-  +  ;/?  =  1, 

(5)  x-/a-  +  y-/h-  =  1,  (e)  the  evolute  (r/x)f  +  (?;;/)!  =  (a-  —  5-)t. 

7.  Find  centers  of  gravity  and  iiionients  of  inertia  al)out  the  axes  in  Ex.  6. 

8.  Find  volumes,  centers  of  gravity,  and  moments  of  inertia  of  the  octants  of 
(a)  xJ  +  yl  +  23  =  fli         (^)  x?  +  z/1  +  zi  =  rn,         (7)  x-  +  //-  +  zt  =  1. 

9.  (a)  The  sum  rif  fi mr  proper  fractions  does  not  exceed  unity  ;  find  the  average 
value  of  their  pmrluct.  (P)  Tlie  same  if  tho  sum  fd  the  sijuares  does  not  exceed 
tmity.    (7)  What  are  the  results  in  the  case  of  k  proper  fractions  ? 

10.  Average  ,—  '</--'''/-  under  the  supposition  ox-  +  by-  =  //. 

11.  Evaluate  the  definite  integral  (15')  by  differentiation  under  the  sign. 

12.  From  (18)  and  1  <   — <  1  +  tt  show  that  the  magnitude  of  D-  log  T  (n) 

is  about  \/n  fnr  large  values  df  n. 

13.  From  Ex.  12.  and  Ex.  23.  p.  70,  show  that  the  error  in  taking 

lotr  r  (n  +  -]     for      f  "      loi:  T  (x)  dx     is  about     log  r  ( 71  +     I  • 

^      \         2'  J„  '^      ^  '  24  71+  12     "^      \         2/ 

14.  Show  that    I  logr(x)(Zx=  I     hig  F  ()i  +  x)  ^Zx  and  hence  compare  (1.5'), 

J  )i  ''  .'0 

(20).  and  Ex.  13  to  .'.^how  that  the  small  quantity  77  is  about  (24  77  +  12)- 1. 

15.  Use  a  four-place  table  to  find  the  logarithms  of  5!  and  10!.  Find  the 
li>garithms  of  the  approximate  values  by  (20).  and  determine  the  percentage  errors. 

16.  Assume  n  =  11  in  (17)  and  evaltiate  the  first  integral.  Take  the  logarithmic 
derivative  of  (20)  to  find  an  approximate  expression  for  T'(n)/T  (n).  and  in  partic- 
ular compute  tile  value  for  71  =  11.  C'ljmbine  the  results  to  find  7  =  0..578.  By  more 
accurate  methods  it  may  be  shown  that  Euler's  Constant  7  =  0. .577. 21 -o. 005  .... 

17.  Integrate  (10')  from  71  to  7i  +  1  to  find  a  definite  integral  for  (15').    Subtract 

Til,  r"  t'""  —  e"^  dcx     ^^  „    , 

the  integrals  and  add     loo-  ,j  — Hence  find 

2     ^  J-..        2         a 


' —    1  r"  r    1        in 

lo-  r  (i()  —  n  (lo-  71  —  1 )  —  1<  1-  ^ '  2  TT  +  ^  lo!,wi  =  I h  -    e"" 

2     "         J-r.  Lc"  —  1       a      2  J 


da 


386  IKTEGKAL  CALCULUS 

18.  Obtain  Stirling' is  approximation.  F  ()i  +  1)  =  Vz7r»?i"e-",  either  by  compar' 
ing  it  with  the  one  already  found  or  by  applying  the  method  of  the  text,  with  the 
substitution  x  =  n  +  \^iy,  to  the  original  form  (1)  of  T  {?i  +  1). 

,«     ^,  ,      .  ^'%Ii.'~^    .     JiTT  .      TT    .      2  TT  .      (?i—  l)7r  n 

19.  The   relation       JT     sn^  —  ==  •'^m  —  sm •  ■  sui  -^ —  = may   be 

1=1  n  n         n  n  2«-i 


»-l  =  (x-  l)TTV/-e      "  y, 


Ini 

(n- 

-ll"' 

('-  " 

e 

obtained  from  the  roots  of  unity  (§  72)  ;  for  x 

.r"-l      ^  =  "-'(1         -""Pi  ^  =  "-ie«        e'    "-  1 

n  =  Inn =     TT     M  —  t'       '   /  TT      = = 

3-ii.r  — 1  /.■  =  i  ;,  =  i      -21        (2?)''-i       2''-i 

150.  The  error  function.  Suppose  that  measurements  to  determine 
the  magnitude  of  a  certain  object  be  made,  and  let  //i^.  //^,.  ■  ■  • ,  ;«„  be  a 
set  of  n  determinations  each  made  independent!}'  of  the  otlier  and  each 
worthy  of  the  same  weight.    Then  the  quantities 

'ii  =  '"1  -  "^  '7.2  =  '"2  -  ^'h  ■■■,  'In  =  w„  -  m, 

Avhich  are  the  differences  between  the  observed  values  and  the  assumed 
value  m,  are  the  errors  committed ;  their  sum  is 

'/l  +  '/2  "I +  '/»   =    ('"1  +    "'-2  +   •  •  •   +    '"„)  —   ''"'• 

It  will  be  taken  as  a  fundamental  axiom  that  on  the  average  the  errors 
in  excess,  the  positive  errors,  arid  the  errors  in  defect,  the  negative 
errors,  are  evenly  balanced  so  that  their  sum  is  zero.  In  other  words  it 
will  be  assumed  that  the  mean  value 

n//i  =  III    +  m.  +  •  •  ■  +  //'„     or     m.  =  —  (ni   +"',  +  •••  +  "'„)   (21) 

1  -  ^^  1  -  .        V         / 

i-;  the  most  probable  value  for  ///  as  determined  from  m ^,  vi ,„  ■  ■  ■ ,  ?>/„. 
Note  that  the  average  value  m  is  that  which  makes  the  sum  of  the 
squares  of  the  errors  a  minimum  ;   lu'nce  the  term  "  least  squares. ■" 

Befon^  any  observations  have  been  taken,  the  cliance  that  any  par- 
ticular error  y  should  be  made  is  0,  and  the  chance,  that  an  error  lie 
within  infinitesimal  limits,  say  l^etween  //  and  y  +  </'/,  is  infinitesimal ; 
let  the  chance  be  assumed  to  l)e  a  function  of  the  size  of  the  error,  and 
write  (f)('/)'If/  as  the  chance  that  an  error  lie  l)etween  y  and  7  +  '/y.  It 
may  be  seen  that  c^c/)  niay  l)e  expected  to  det-rease  as  y  incrtnises  ;  for, 
under  the  reasonable  hypothesis  that  an  observer  is  not  so  likely  to  l.e 
far  wrong  as  to  lie  sonu'where  near  right,  the  chance  of  making  an 
error  ])etweeii  S.O  and  S.l  Avould  be  less  than  that  of  making  an  eri'or 
between  1.0  and  1.1.  The  function  ^(y)  is  called  the  error  function. 
It  will  l)e  said  that  the  chance  of  making  an  eri-or  y-  is  <^(y,);  to  put  it 
more  precisely,  this  means  simply  that  ^('y,)"'y  is  the  chance  of  making 
an  error  which  lies  ijetween  y,-  and  y,.  +  r/y. 


FUNCTION'S  DEFINE!)  BY  INTEGRALS  387 

It  is  a  fundanit'iital  principle  of  the  tlieoi'v  of  cliaiice  that  tlie 
chance  that  several  iude})eiident  events  take  ])lace  is  tlit;  product  of 
the  chances  for  each  separate  event.  The  })rol)aljility,  then,  that  the 
errors  q^,  q^,---,  ^„  be  made  is  the  product 

<^('/i)  <^('Z2)  •  •  •  <^(V«)  =  <^('«i  -  r>i)  <i>(m,  -  rn)  •  •  .ci>(m,^  -  m).   (22) 

The  fundamental  axiom  (21)  is  that  this  probability  is  a  maximum 
when  m  is  the  arithmetic  mean  of  the  measurements  m.^,  m,^,---,  7//,,: 
for  the  errors,  measured  from  the  mean  value,  are  on  the  whole  less 
than  if  measured  from  some  other  \-alue.*  If  the  probability  is  a  maxi- 
mum, so  is  its  logarithm;  and  the  derivative  of  the  logarithm  of  (22) 
with  respect  to  11/  is 

^'(Wl  —  I)))         4>'{lll.,  —  lit)  4>'('"»  —  w)  ^ 

<i>{rn^  —  III)        4>(j"o  ~~  "0  4'(,"',i  ~  "0 

when  +'/.,+  ■••  +  '/„  =  ('''1  —  "0  +  ('"2  ~  "0  +  "  ■  •  +  ("In  —  '"0  =  0- 
It  remains  to  determine  <^  from  these  relations. 

For  brevity  let  F(<f)  be  the  function  F  =  <^' '<A  wliic^h  is  the  ratio 
of  <^'('/)  to  (f>(q)-    Then  the  conditions  become 

F('I,)+F{q.^  +  ...  +  F(q,^)  =  0      when      y^  +  y.^  +  . . .  +  .y,,  =  0. 
In  particular  if  there  are  only  two  observations,  then 

F(q^)  +  F(q,^  =  0      and      y^  +  y,  =  0      or      q,  =  -q,. 
Then  /-Yy^)  +  Fr- y,)  =  0     or     F(-y)=-F(y). 

Kext  if  there  are  three  oljservations.  the  results  are 

^(7i) +  /«' (//,)  + i-^'/,)  =  0     and      y, +  y,  +  y,  =  0. 
Hence         F(q;)  +  F('/,)  =  -  /' (y,)  =  /■(  -  '/,)  =  ^'('/,  +  '/,)• 
Now  from  ^(.•'')  +  ^(f/)  =  ^'^^■''  +  //' 

the  function  F  may  be  determined  (Ex.  9,  p.  IT))  as  F{.r)  =  Cx.    Then 

9  i'l) 
and  <^(y)  =  e'^^'  +  ^=G'r*'^''. 

This  determination  of  <^  contains  two  arbitrary  constants  which  may 
be  further  determined.  In  the  first  ]>lace,  note  that  C  is  negative,  for 
(t>(q)  decreases  as  q  increases.   Let  -i-  C  =  —  A-".    In  the  second  place,  the 

*  The  derivation  of  the  expression  for  4>  is  pliysical  rather  tlian  loijical  in  its  arizu- 
ment.  'riie  real  justitieation  or  proof  of  the  validity  of  the  expression  obtained  is  a  pos- 
teriori and  depends  on  the  experience  that  in  practice  errors  do  follow  the  law  (24). 


388  INTEGRAL  CALCULUS 

error  y  must  lie  within  the  interval  —  cc  <  y  <  +  x  which  comprises 
all  possible  values.    Hence 

r\{q)dq  =  l,  of     \-^'rdq=l.  (23) 

*y  —  cc  •  U  —  -Ji 

For  the  chance  that  an  error  lie  between  q  and  q  +  dq  is  ^'/y,  and  if 
an  interval  a^q^h  be  given,  the  chance  of  an  error  in  it  is 

^  <^  (7)  dq     or,  better,     lini  ^  (^  (y)  ^Ay  =  /    <^  (7)  ^A/, 

and  finally  the  chance  that  —  cc  <  y  <  +  ^  re})resents  a  certainty  and 
is  denoted  by  1.  The  integral  (23)  may  be  evaluated  (§  143j.  Then 
a  Vtt/A-  =  1  and  r;  =  /.■/ Vtt.    Hence  * 

<^(y)  =  ^e-^'V  (24) 

The  remaining  constant  /.•  is  essential ;  it  measures  the  accuracy  of 
the  observer.  If  k  is  large,  the  function  <^(y)  falls  very  rapidly  from 
the  large  value  A-/ Vtt  for  y  =  0  to  very  small  values,  and  it  ai)i)ears 
that  the  observer  is  far  more  likely  to  make  a  small  error  than  a  large 
one;  but  if  k  is  small,  the  function  c^  falls  very  slowly  from  its  value 
A'/ VTT  for  y  =  0  and  denotes  that  the  observer  is  almost  as  likely  to 
make  reasonably  large  errors  as  small  ones. 

151.  If  only  the  numerical  value  be  considered,  the  probability  that 
the  error  lie  numerically  between  y  and  y  -f-  '^q  i~' 

2  A-  2  A'    r^ 

—,— e-''''rdq,     and     — r^  \     <'-^''rdq 
Vtt  VttJ, 

is  the  chance  that  an  error  be  numerically  less  than  ^.    Xow 

2k     r^  2      r'^ 

^(0  =  -/-         c-^V,Ay  =  — .  ,.-.>V,,.  (25) 


is  a  function  defined  by  an  integral  with  a  variable  ui)})er  limit,  and  the 
problem  of  computing  the  value  of  the  function  for  any  given  vaku;  of  ^ 
reduces  to  the  problem  of  conqjuting  the  integral.  The  integrand  may 
be  expanded  by  ^laclaurin's  Formula 

„       ,  ,       3'*       ./•*=       .r«       :,^i'-e^  _^        ^        ^ 

e-^^  =  l-.r  +  ~  -  .yi  +4-,  -  -^T-'  ^  <  ^  <  l- 


£ 


*  The  reader  may  now  verify  the  fact  tlwit,  with  <t>  as  in  (iM),  tlie  itroduct  ('2'2)  is  a 
maximum  if  the  sum  of  the  squares  nf  the  errors  is  a  iniuimum  as  (h'liiaiKh'd  l)y  Cil). 


FUNCTIONS  DEFINED  BY  INTEGRALS  389 

For  small  values  of  x  this  series  is  satisfactory  ;  for  cc  S  |-  it  will  be 
accurate  to  five  decimals. 

The  -probable  error  is  the  technical  term  used  to  denote  that  error  ^ 
which  makes  \p{^)  =  \;  that  is,  the  error  such  that  the  chance  of  a 
smaller  error  is  |-  and  the  chance  of  a  larger  error  is  also  \.  This  is 
found  by  solving  for  x  the  equation 


V^   1 


1  r^  x^     x^     x' 

-  =  0  .44311  =  /     e~^dx  =  x--  +  ™-  —  +  7 
2  J^  3       10      42       . 


X 

2      2  ^^~Jo    '        "  3    '   10      42   '   2T()' 

The  first  term  alone  indicates  that  the  root  is  near  x  =  .45,  and  a  trial 
with  the  first  three  terms  in  the  series  indicates  the  root  as  between 
X  =  AT  and  x  —■  .48.  With  such  a  close  approximation  it  is  easy  to  fix 
the  root  to  four  places  as 

X  =  H  =  0.4769     or     ^  =  0.4769  k-\  (27) 

That  the  probable  error  should  depend  on  k  is  obvious. 

For  large  values  of  x  =  Ji$  the  method  of  expansion  by  Maclaurin's 
Formula  is  a  very  poor  one  for  calculating  i/'(^);  too  many  terms  are 
required.  It  is  therefore  important  to  obtain  an  expansion  according 
to  descending jJoivers  of  x.    Now 

Js-'^'dx  =  I      e-^'dx  —  I      e-^-dx  =  -  v  tt  —   /      e-'^'dx 
U  t/O  tJ  X  "'  Jx 

and  e-'-dx=         -xe~^-dx=    — -rr—       —7;    I      —^, 

/  I      X  2x  2  }         x^ 

The  limits  may  be  substituted  in  the  first  term  and  the  method  of  in- 
tegration 1)}'  parts  may  be  a})plied  again.    Thus 

r"       .,         e-^V.         1\       1-3    C^  er'^Wx 

_  ^-j^  /      j_     i^\  _  1  ■  3  ■  5  r*  e-^\Jx 

~  2x\       ■2.r-^^2-./'7  2^     J,,         x^     ' 

and  so  on  indefinitely.    It  should  be  noticed,  however,  that  the  term 

^      l-3-5---(2?i-l)  «--  ,. 

T  =  ■ ^^^ '-  j^  diverges  as  n  =  cc. 

In  fact  although  the  denominator  is  midtiplied  by  2x'^  at  each  ste}),  the 
numerator  is  multiplied  by  2  71  —  1,  and  hence  after  the  integrations  by 
})arts  have  been  aj)])lied  so  many  times  that  71  >  x^  the  terms  in  the 
parenthesis  begin  to  increase.  It  is  worse  than  useless  to  carry  the 
integrations  further.    The  integral  which  remains  is  (Ex.  5,  p.  29) 


390  INTEGRAL  CALCULUS 

l-3-5---(2n  +  l)   r"  «-"V.y        1.3.o-.-(2?^-l) 

^ ^-^   '       <  ,..,  ,,  .,..  .. -fi'""'  <  T. 


2n+l 


T5^ 


Thus  the  integral  is  less  than  the  last  term  of  the  parenthesis,  and  it 
is  possible  to  write  the  asymjjtot'iG  series 

r""      2,         1     /-       '^~"V.         1         1-3       1-3-0  \ 

i  ^"^^^•'^  =  2^-ir7(i-T:^  +  2^""2^  +  --V      ^-^^ 

with  the  assurance  that  tJie  value  ohtained  by  rising  tlie  series  vj ill  differ 
fritrii  tJie  true  value  by  less  than  the  last  terinwhh-h  Is  used  m  the  series. 
This  kind  of  series  is  of  frequent  occurrence. 

In  addition  to  the  probable  error,  the  areraije  nunwrlcd  error  and  the 
mean  square  error,  that  is,  the  average  of  the  square  of  the  error,  are 
important.  In  finding  the  averages  the  probability  <^  (7)  dq  may  be  taken 
as  the  weight ;  in  fact  the  probability  is  in  a  certain  sense  the  sinqJest 
Aveight  because  the  sum  of  the  weights,  that  is,  the  sum  of  the  ]jrob- 
abilities,  is  1  if  an  average  over  the  whole  range  of  possible  values  is 
desired.    Tor  the  average  numerical  error  and  mean  square  error 

,—       21:     r^       ,.,,,  1  0.5(>43 

-,        2k     C   .,,,,,  1  /=      0.7071  "^ 

"'  =  ^1.    *-«"*■'■'"■''' =  2P'         ^V  =  -l— 

It  is  seen  that  the  average  error  is  greater  than  the  probable  error,  and 
that  the  square  root  of  the  mean  square  error  is  still  larger.  In  the 
case  of  a  given  set  of  n  ol)ser\'ations  the  averages  niay  actually  be 
computed  as 

\q\  —  — ■  —  — J=  ?  A  —  =: — ^  > 

'''  /.■  Vtt  j^l  Vtt 

-  'l^+'^'   +   ■■■    +   'ln  1  ;  1 


\ 


?V5 


]Vroreover, 


It  cannot  be  expected  that  tlie  two  values  of  /.■  thus  fomid  will  be  pre- 
cisely equal  or  that  the  last  relation  will  be  exactly  fulfilled:  but  so 
well  does  the  theory  of  errors  represent  what  actually  arises  in  prac- 
tice that  unless  the  two  values  of  k  are  nearly  ecjual  and  the  relation 
nearly  satisfic^l  there  are  fair  reasons  for  sus})ecting  tliat  the  observa- 
tions are  not  bona  fide. 

152.    Consider  the  question   of  tlie   a])]»lication   of  these  tlieories  to 
the   errors   made    in    rifie    pi-actiee    on   a   tarLret.      Here    there    are    two 


FUNCTIONS  dp:fixed  by  integrals  391 

errors,  one  due  to  the  fact  that  the  shots  may  fall  to  the  right  or  left 
of  the  central  vertical,  tlie  other  to  their  falling  above  or  below  the 
central  horizontal.  In  other  words,  each  of  the  coordinates  (.r,  y)  of 
the  position  of  a  shot  Avill  be  regarded  as  subject  to  the  law  of  errors 
independently  of  the  other.    Then 

h  ,  .  A-'  „  „  l:k'       r.  „     ,„ ., 

Vtt  Vtt  '^ 

will  be  the  probabilities  that  a  shot  fall  in  the  vertical  strip  between 
X  and  X  +  (hr,  in  the  horizontal  strip  between  y  and  y  +  dy,  or  in  the 
small  rectangle  common  to  the  two  strips.  ^Moreover  it  will  be  assumed 
that  the  accuracy  is  the  same  with  respect  to  horizontal  and  vertical 
deviations,  so  that  k  —  k'. 

The.se  a.ssuinptions  may  appear  too  special  to  be  reasonable.  In  particular  it 
might  seem  as  thouuh  the  accuracies  in  the  two  directions  wonld  be  very  dil^erent, 
owing  to  the  possibility  that  the  marksman's  aim  sliould  tremble  more  to  the  right 
and  left  than  up  and  down,  or  vice  ver.sa,  .so  that  k  :^  k'.  In  this  case  the  shots  would 
not  tend  to  lie  at  equal  distances  in  all  directions  from  the  center  of  the  target, 
but  would  dispose  themselves  in  an  elliptical  fashion.  Moreover  as  the  .shooting  is 
done  from  the  right  shoulder  it  might  seem  as  though  tliere  would  be  .some  inclined 
line  through  the  center  of  the  target  along  which  the  accuracy  would  be  least,  and 
a  line  perpendicular  to  it  along  which  the  accuracy  would  be  greatest,  .so  that  the 
disposition  of  the  .shots  would  not  only  be  elliptical  but  inclined.  To  cover  this 
general  assumption  the  probability  would  be  taken  as 

Q^-k'-x---2Kxi,-k'->r,ij-j}ij^     with     G   C         i  t-f^'-'='-->^-'-y-'^''-'rdxdy  =  1 
as  the  condition  that  the  shots  lie  somewhere.    See  the  exercises  below. 

With  the  special  assumptions,  it  is  best  to  transform  to  polar  coor- 
dinates. The  important  (juantities  to  determine  are  the  average  distance 
of  the  shots  from  the  center,  the  nuniu  square  distance,  the  j^roljuble 
distance,  and  the  nujst  probable  distauc.-e.  It  is  necessary  to  distinguish 
carefully  between  the  probable  distance,  Avhich  is  by  definition  the  dis- 
tance such  that  half  the  shots  fall  nearer  the  center  and  half  fall  farther 
away,  and  the  most  probable  distance,  which  by  definition  is  that  dis- 
tance which  occurs  most  frequently,  that  is,  the  distance  of  the  ring 
between  /■  aiul  /'  +  "'/"  iu  whieh  most  shots  fall. 

The  probability  that  the  shot  lies  in  the  element  rdrdf^  is 

—  e- ^'''"■/y//y/</),     and     2  Irti- '"'''' rdr, 

IT 

oV)tained  by  integniting  with  respect  to  ^,  is  the  ])rol)ability  that  the 
shot  lies  ill  tlie  ring  from  /■  to  /•  +  '//■.    The  mn^^f  jii'dlmldf  distance  r ^^  is 


392  INTEGRAL  CALCULUS 

that  wliicli  makes  this  a  maximum,  that  is, 

'//,-..  ..  1  0.7071 

_(.-,,,,  =  0     o,-     ,,  =  -^  =  --^.  (30) 

The  mean  distance  ami  tlie  mean  square  distance  are  respectively 

V^  _       0.SS62 


2  lr,>-  k-r"-j.-\/,.  = 

2  Ire-  ^  '  r'\l)'  =  -j-; '  V  /-  =  — 


(30') 


(30") 


Tlie  pmhahle  distance  r^  is  found  b}-  solving  the  e(juation 

~  =  I       2  /.•  ,->-  >  ,'ilr  =  1  -  f-'-  '5  ,  /-J  = -—  =  — 

Hence  z'^,  <  r^  <  e  <  V /•-. 

The  chief  importance  of  these  considerations  lies  in  the  fact  tliat, 
ou'ing  to  ]\Laxwell's  assumption,  analogous  considerations  may  he  applied 
to  the  velocities  of  the  molecules  of  a  gas.  Let  a.  r.  v  Ite  the  c()m})0- 
nent  velocities  of  a  molecule  in  three  jjerpendicular  directions  s(j  that 
r  =  {a-  -\-  r-  -j-  a--)-  is  the  actual  velocity.  The  assum^ition  is  mad''  that 
tlie  individual  components  a.  r.  w  obey  the  law  of  errors.  The  proba- 
bility that  the  components  lie  between  the  respective  limits  k  and  u  -f-  du, 
r  and  r  +  "''",  "'  lUid  (C  -\-  dw  is 

,.-'^>^-'^'---iy'>'-,hn]ri]ir^      and      ^  p-'- '■■-!'- sin  0<n\/Od(f) 


TT  V  TT  TT  V  TT 

is  the  corresponding  expression  in  |)olar  coordinates.  There  will  then 
be  a  most  probable,  a  proljable,  a  mean,  and  a  mean  S(]uare  velocitv. 
Of  these,  the  last  corres})onds  to  the  mean  kinetic  energy  and  is  subject 
to  measurement. 

EXERCISES 

1.  Tf /,•  "  ().()447').  lind  to  three  i>lacf>  the  prnhahility  nf  an  error  ^  <  12. 

2.  Coiiipiin-    r   i:--'-\U  to  three  phices  for  {a)  x  =  0.2.  (pi)  j-  =  0.8. 

3.  State  how  many  terms  of  (28)  should  be  taken  to  obtain  the  be.st  value  for 
the  intej,a-al  to  x  =  2  and  oljtaiii  that  value. 

4.  IIow  aecurately  will  (28)  determine    /     c- -'-dx  —  I  \^  ?    Compute. 

5.  Obtain  these  asymptotic  expansions  and  extend  them  to  tind  the  ireneral  law. 
Show  tiiat  tlie  erroi-  introihiced  liy  omittiiiLr  tlie  integral  is  h-ss  than  tiie  hist  term 
retained  in  tlie  series.  Show  further  that  the  general  tei'ui  diver;i-es  when  n  be- 
eomes  infinite. 


FUXCTIOXS  ])EFIXED  UY  IXTEGKALS  393 

,    ,     C         ,,         1       TT      .sinx-       cosx^       1-3    r '^  ilx 

(a)    I     cos  j-cZx  =  "-»-! ^ I      t-'osx-  —  , 

,   ,    r''  .      1,        1       TT      cosx-      siiix-      1-3  /'^  .      ,  t/x 

(a)        sill  x'-ax  =  -  -V 1 I     Hiiix-  —  , 

^^'  J(>  ■2\-l  2x  2-^x3  -I^  Jx  x4 

,    ,    r^amx^        .  ,^.     /"^/sinxX^,        , 

(7)    I      cZx,  X  lai-rje,         (5)    |        )  dx,  x  large. 

./o       X  '  Jo     \   X    / 

6.  (a)  Find  the  value  of  the  average  of  any  odd  power  2  n  +  1  of  the  error ; 
(/S)  also  for  the  average  of  any  even  power ;  {7)  also  for  any  power. 

7.  The  observations  195,  225*,  100.  210,  205,  180*,  170*,  190.  200.  210,  210,  220*, 
175*,  192  were  obtained  for  deflections  of  a  galvanometer.  Compute  k  from  the 
mean  error  and  mean  square  error  and  compare  the  results.  Suppose  the  observa- 
tions marked  *,  which  show  great  deviations,  were  discarded  ;  compute  A:  Ijy  the 
two  methods  and  note  whether  the  agreement  is  so  good. 

8.  Find  the  average  value  of  the  product  q(i'  of  two  errors  selected  at  random 
and  the  average  of  the  product  |ry|  •  |^']  of  numerical  values. 

,  ,  .  ,     .  .        .  T,         1      -,,         1.0875 

9.  Show  that  the  various  velocities   for   a   gas   are    T  ^j  =     ,    If  = , 

^_     2      _  1.1284        /^_    V^    _  1.2247  ''  ^ 

~Vxt"       ^      '  V2A:  ^ 

10.  For  oxygen  (at  0^  C.  and  7Gcm.  Hg.)  the  square  root  of  the  mean  S(iuare 
velocity  is  402.2  meters  per  second.  Find  k  and  show  that  only  about  13  or  14 
molecules  to  the  thousand  are  moving  as  .slow  as  100  m. /sec.  What  .speed  is  most 
probable  ? 

11.  Under  the  general  assumption  of  ellipticity  and  inclination  in  the  distri- 
buticiii  of  the  shots  show  that  the  area  of  the  ellipse  k-x'~  +  2  Xx/y  +  k'-y-  =  II  is 
ttII  {k-k"-  —  X-)~  ■-.  and  the  probability  may  be  written  Ge-  "TT{k-k"-  —  X-)"  -dll. 

1 


12.  From  Ex.  11  establish  the  relations         (a)   G  =-  Vk-k'-  -  X-, 

TT 

13)  x^  = ' ,         (7)  ?/-  =  ~ ,         (5)  x7/  =  ~^ -. 

13.  Find  H,,.  11^  =  0.093,  U.  IT-  in  the  above  problem. 

14.  Take  20  measurements  oi  s(jme  ol)ject.  Determine  A:  by  the  two  nieth(jds 
and  ciiinijare  the  results.    Test  other  points  of  the  theory. 

153.  Bessel  functions.  The  use  of  a  detinite  integral  to  detine  func- 
tions w"liich  satisfy  a  given  differential  ecjuation  may  be  illustrated  by 
the  treatment  of  .ry"  +  {'In  +  1)//'  -f  a-y  =  0,  which  at  tlie  same  time 
will  afford  a  new  investigation  of  some  functions  which  have  ])re- 
viously  been  Itrieily  discussed  ('§§107-108).  To  obtain  a  solution  of 
tliis  e(juation,  or  of  any  equation,  in  the  form  of  a  definite  integral,  souu^ 
special  tyi)e  of  integrand  is  assumed  in  pai-t  and  tlie  remainder  of  the 


394  INTEGRAL  CALCULUS 

integrand  and  the  limits  for  the  integral  are  then  determined  so  that 
the  equation  is  satisfied.    In  this  case  try  the  form 

y {:!■)=   I  e'^^'Tdt,  y'  =  i  if<''''TtJt,  y"  =  I  —  t^e^^'Tdt, 

where  T  is  a  function  of  f,  and  the  derivatives  are  found  by  differen- 
tiating under  the  sign.  Integrate  y  and  y"  by  parts  and  substitute  in 
the  equation.    Then 

(1  -  f)  Te'^'^  -  C e'^-'lT^l  -  f) -\- (2  n  -  l)t7yh  =  0, 

wliere  the  bracket  after  the  first  term  means  that  tlie  difference  of  the 
values  for  the  u})per  and  lower  limit  of  the  integral  are  to  be  taken ; 
these  limits  and  the  form  of  T  remain  to  be  determined  so  that  the 
expression  shall  really  be  zero. 

The  integral  may  be  made  to  vanish  by  so  choosing  T  that  the 
bracket  vanishes  ;  this  calls  for  the  integration  of  a  simple  differential 
equation.    The  result  then  is 

r  =  (i-  t-y  -i,      (1  -  ty  +  ^«''']  =  0. 

The  integral  vanishes,  and  the  integrated  term  will  vanish  provided 
t  =  ±  1  or  e"'*  =  0.  If  X  be  assumed  to  be  real  and  positive,  the  expo- 
nential will  a2)proa(;li  0  when  t  =  1  -\-  iK  and  K  becomes  infinite.    Hence 

y(x)=  e'-'-'(l-f-y-^dt     and     ^  (,'■)=  c'-''(l  -  t-f- ^dt    (^M) 

^'-i  J+i 

are  solutions  of  the  differential  e(|uation.  In  the  first  the  integral  is  an 
infinite  integral  Avhen  ?4  <  +  ]L  and  fails  to  converge  Avhen  n  ^  —  \. 
Tlic  solution  is  therefore  defined  oidy  when  ?i  >  —  },.  The  second  in- 
tegral is  always  an  infinite  integral  because  one  limit  is  infinite.  The 
examination  of  the  integrals  for  uniformity  is  found  below. 

r "  ■  - 1        ' 

Consider   j       (,'-'''(l  —  t-)"     idt  with  n  <  I  so  tliat  the  iiitcyral  is  infinite. 

r      c>''{\  -  i-f-  '-  dt=  C      (1  -  i-j"  -  I  cuaxtdt  +  '  f      (1  -  '"/'  ~  ^  sinxWi. 

From  considerations  of  symmetry  tlie  second  intei^ral  vanislies.    Then 

r'  c>-<^'{l-t-)"-ku\  =  \  r'  n-t~)"~'-coiixtdt\^   f     (l-r-)"-itZL 

'J'his  hist  intenral  with  ai)osiiiv('  integrand  convciryes  wlien  ?i  >  —  l,  and  'lence  the 
giviai  inteu'ral  converues  unit'oiinly  for  all  values  of  x  and  delines  a  '.'ontinuous 
function.    Tiie  su(;cessive  ditlerenliations  under  the  si^n  i;'ive  the  I'esults 


FUXCTIOXS  DEFINED  BY  INTEGKALS  395 


-  I       {l-t-)"'^t  sin  x^(Zi,         -   /       (1  -  t-)"  "  2  f-  COS 


xtdt. 


These  integrals  also  converge  uniformly,  and  hence  the  differentiations  were  justi- 
fiable.  The  second  integral  (31)  may  be  written  with  t  =  1  +  iii,  as 


\     J  u  =  0  «^  0 


du. 


This  integral  converges  for  all  values  of  x  >  0  and  n  >  —  I.  Hence  the  ;  iven  inte- 
gral converges  uniformly  for  all  values  of  x  ^  Xq  >  0,  and  defines  a  continuous 
function  ;  when  x  =  0  it  is  readily  seen  that  the  integral  diverges  and  could  not 
define  a  continuous  function.    It  is  easy  to  justify  the  differentiations  as  before. 

The  first  form  of  the  solution  may  be  expanded  in  series. 

X-l  r>  +1 

e''-'(l  -  f-y  -  '^  (If  =  (1  -  t-y  ~  ^'  cos  xtdt 

=  2   r  (1  -  ^')""  -  cos  xtdt  (32) 

XI  1  /         r-f^       r*;'*       r'^f^  r^f^\ 

a-'T-^(i-'2^  +  |T-'6T  +  «8T}"'  o<l«l<i- 

The  expansion  may  be  carried  to  as  many  terms  as  desired.  Each  of 
the  terms  separately  may  be  integrated  by  B-  or  T-functions. 

_     X-  ^T  (??-  +  ^ )  r  (k  -f-  ^)  x''^T(n-{-  ^)  Vtt 

~  r(2  A-  +  l)T{n  +  k  +  r)~  2-'T{l-  -f  1)  r(/i  +  k  +  1)  ' 

is  then  taken  as  the  definition  of  the  special  function  J„Q'')i  where  the 
expansion  may  be  carried  as  far  as  desired,  with  the  coefficient  6  for 
the  last  term.  If  n  is  an  integer,  the  F-functions  may  be  written  as 
factorials. 

154.   The  second  solution  of  the  differential  equation,  namely 


^  (•'■)  =  Z/i  (-r)  +  i!/^-)  =  J '  '  ' '  -  2  .^'-'(1  -  fy  -  i  dt, 


(31') 


where  the  coefficient  —  2  has  been  inserted  for  convenience,  is  for  some 
jiurposes  more  useful  than  the  first.  It  is  complex,  and,  as  the  equation 
is  real  and  .r  is  taken  as  real,  it  affords  two  solutions,  namely  its  real  part 
and  its  ])ure  imaginary  part,  each  of  which  must  satisfy  the  equation.  As 
v/(,r)   converges  for  ,r  =  0  and  ':(x)  diverges  for  a'  =  0,  so  that  i/^(x)  or 


396 


INTEGRAL  CALCULUS 


//.,(•"'■)  diverges,  it  follows  that  //(./■)  and  ?/i(.>")  or  //(,'■)  and  ,'/„(■'•)  must  Ije 
independent;  and  as  the  equation  can  have  but  two  inde])endent  solu- 
tions, one  of  the  pairs  of  solutions  must  constitute  a  com- 

s 
plete  solution.    It  Avill  now  be  shown  that  ///.'■)  =  l/O'') 

and  that  Ai/(:r')  +  J\'/.,(-'')  is  therefore  the  complete  solu- 
tion of  xij"  +  (2  ?i  +  1)  //'  -f  ./•//  =  0. 

Consider  the  line  integi'al  around  the  contour  0,  1  —  e, 
1  +  ei,  1  +  cc  i,  X  /,  0,  or  OPQliS.  As  the  integrand  has  a 
continuous  derivative  at  every  point  on  or  Avithin  tlic 
contour,  the  integral  is  zci'O  (§  124).  The  integrals  along 
the  little  quadrant  ]'Q  and  the  unit  line  US  at  infinity  may  be  made  as 
small  as  desired  by  taking  the  quadrant  small  enough  and  tlie  line  far 
enough  away.  The  integral  along  SO  is  pure  imaginary,  namely,  witli 
t  =  bi, 

J  so  Jo 

The  integral  along  OP  is  coni})lex,  namely 

-2  «'-'(!-  f-f-^-dt 

=  —  '2  \      '  1  —  ''■)"  ~  '  f'os  :rf,lf  -  2  !  I     (1  -  f-)" "  3  sin  crff/f. 

(1  _  f-)" -  2  eos  .vTdt  -  2  ;  I      (1  —  f- )" "  2  sin  xtdf  4-  l^ 
+  I      -  2  <■''■' (I  -  /■-/'-  l/^  j^i^j^'2>   i    <■-•■••{  1  +  ii-f-  'r//^, 

J  q  J  I) 

where  t,^  and  ^.,  are  small.  Equate  real  and  imaginary  parts  to  zero 
sejiarately  after  taking  the  limit. 


/ 

J  oi 


1   —  f- f      -  cos  .i-filt  =  //(./•)  = 


^1' 

2  r  (i-f-y-isi 


■i: 


2r''',  1    -/-)' 


\\\j-t'lf  —  2 


--('1  +  v-)'     Ijl 


■I 


2  >-'■■'{  \  ~f- 


//,'•'•>. 


df  =  uj.r). 


The  signs  /^and  ^'  are  used  to  denote  respectively  real  and  imaginary 
Ijarts.  Tlie  identity  of  //(.'■)  and  //,(.'■)  is  establislied  and  the  new  solu- 
tion //.,(.'•)  is  founil  as  a  ditfei-enee  of  two  integrals. 


FUNCTIONS   J)EFIXED  1'>V   INTEGRALS 


3U7 


It  is  now  possible  to  obtain  the  iin})ortaut  expansion  of  the  soluticjns 
//(.r)  and  i/.,(->')  in  (/fsmu/hif/  })0\vei's  of  .'•.    For 

Since  x  ^  0,  the  transformation  ux  =  r  is  permissible  and  gives 


(11  -  A)r". 


2\i2xy 


-  —  u 


The  expansion  by  the  binomial  tlieorem  may  be  carried  as  far  as  de- 
sired ;  but  as  the  integration  is  subsecpiently  to  be  performed,  the 
values  of  r  must  l)e  allowed  a  range  from  0  to  x  and  the  use  of 
Taylor's  Formula  with  a  remainder  is  re(piired  —  the  series  Avould  not 
converge.    The  result  of  the  integration  is 


z(x)  =  2"+h:-"-ir(?i+  l)e^    ('"-D'-^j^^, 


where 


Q  (■'■) 


(n- 


)+iQ(x)l  (34) 


:M(2xf 


+ 


'^■''-^  2\i2xf         "^  ^l{2xf 

Take  real  and  imaginary  parts  and  divide  l)y  2"  x- " -w  ttT  {  n  -\-  ),)•    Then 


Jjx)  = 
K  „(■'■)  = 


2 

TT.r 


P  ( .,. )  cos  I  X  -  I  »  +  ;^ )  ^  j  -  (^  (./• )  sni 
1 


1\  TT 


(<?('./•)  cos  l.r-  hi  + 


+  7'(y)sni 


n  + 


IXtt 


ai'C  two  independent  fJessel  functions  which  satisfy  the  ecpiation  (oo) 
of  §  107.  If  n  +  i-  is  iin  integei',  P  and  (I  terminate  and  the  solutions 
are  expressed  in  terms  of  elementary  functions  (^lOS);  Init  if  7i  +  I 
is  not  an  integer,  /'  and  Q  ai'e  mei-eiy  asym})totic  ex})ressions  wliich  do 
not  terminate  of  tliemselves,  l)ut  nnist  be  cut  short  with  a  remainder 
term  l)ecause  of  their  tendency  to  diverge  after  a  certain  point:  for 
tolerably  large  values  of  ./■  and  small  values  of  7i.  the  values  of  ./,/./") 
and  KJx)  may,  however,  be  computed  with  great  accuracy  by  using 
the  first  few  terms  of  P  and  Q. 


398  INTEGKAL  CALCULUS 

The  integration  to  find  P  and  Q  offers  no  particular  difficulty. 

f"e-  'u"  -  2  +  *cZu  =  r(n  +  i  +  ^)  =  (n  +  k-  i)(n  +  A;  -?)•••  (n  +  i)  T{n  +  h). 
Jo 

The  factors  previous  to  r  (n  +  |)  combine  with  ji  —  ^,  n  —  |,  •  •  • ,  »  —  A;  +  J,  which 
occur  in  the  A:th  term  of  the  binomial  expansion  and  irive  the  numerators  of  the 
terms  in  P  and  Q.  The  remainder  term  must,  however,  be  discussed.  The  integral 
form  (p.  57)  will  be  used. 

-——^fao(v-t)dt, 

Let  it  be  .supposed  that  the  expan.sion  has  been  carried  so  far  that  n  —  k  —  I  <0. 

Then  (1  +  vi/2x)"~^'~  -  is  numerically  greatest  when  r  =  0  and  is  then  etpial  to  1. 

Hence 

IB  l<   f      t'-'      \(n-l)---(n-k  +  l)\^^^v'^\(n-l)...(n-k+^)\ 
I    ^1      Jo    (A;_l)!  (•2/)^-  k'.  (2/)i- 


and 


!  X  '' "'  '  '''■''  I  < FFT? ^  ("  +  2)  ■ 


It  therefore  appears  that  when  fc  >  ?!  —  I  the  error  made  in  neglecting  the  remain- 
der is  less  than  the  last  term  kept,  and  for  the  maxinuim  accuracy  the  series  for 
P  +  iQ  should  be  broken  off  between  the  lea.st  term  and  the  term  just  following. 

EXERCISES 

1.  Solve  xy"  -\-  {2  n  +  1)  y'  —  xy  =  0  by  trying  Te^'  as  integrand. 

A  f     (1  -  f^)" - ^e'Ult  +  B  C     (<2  _  i)« -ie-r'dt,         x>0,         n>-  i. 

2.  Expand  the  first  solutirm  in  Kx.  1  into  series;  compare  with  y(Lr)  above. 

3.  Try  7(1  -  tx)'"  on  x{l-  x)  y"  +  [y  -  (a  +  ^  +  1)  x]  y'  -  cx^y  =  0. 

One  solution  is     f  <3-i(l  _  <)7-3-i(l  -  i.f)-«d^,         ^  >  0.         7  > /S,         |x|<l. 

4.  Expand  the  .solution  in  Ex.  -3  into  the  series,  called  hyi:)ergeometric, 

T,        f^)3  a  (ex  +  1) 

iJ(/3,  7-/3)    1  +  -^^+      \.^    ' 
\_         1-7  1-  2  7  i 


1),3(^  +  1)^, 


7 (7  +  1) 

a(a+  l)(ar  +  2)/3(;3  +  l)(3+  2)  .^„       ^  _  T 

1.2.37(7  +  l)(7  +  :i)  ■'^        '"'J' 

5.   EstaVilish  tliese  results  for  Bcssel's  ./-functions  : 

X^  C  ^ 

(<t)  J,t(x)  =  I      sin-"  (p  cos  {x  cos0)(Z0,         n  >  —  h 

(/3)  -/„(.r)  = ^ f     sin-"  0cris  (,r  cos0)rZ0,         >«  =  0,  1.  2.  3  • 

TT  3  •  o-  •  ■  (2  ?i  —  1)  Jn 


FUNCTIONS  DEFINED  EV   l^sTEGKALKS  o99 

6.  Show  -   (      co.s  (?i0  —  X  sin  cp)  d(p  satisfies 

?/       /,       n^\  sin  7?7r /I        n\ 

/'  +  -+!--,   y  = ,  • 

X  \  X-/  TT         \X          X^/ 

7.  Find  the  CMiuation  of  the  sf  cc;)  J  order  satisfied  by   /     (1  — <-)"     ^sinxid^. 

rpi  rpG  7»8  X^^ 

8.  Show  J„(2  x\  =  1  -  X-  + ; +  — ^— ^— -  +  •  •  •  . 

9.  Compute  ./^(l)  =  0.7652  ;  Jq{2)  =  0.2239  ;  J^i^AOo)  =  0.0000. 

10.  Prove,  from  tlie  integrals,  J,){.r.)  =  —  '/j(x)  and  [x- "-/„]'  =  —  -r->'J„  ^i . 

11.  Show  that  four  terms  in  tlie  asymptotic  expansion  of  P  +  i(^  wlien  n  =  0 
give  the  best  result  when  x  =  2  and  tliat  the  error  may  be  about  0.002. 

12.  From  the  asymptotic  expansions  compute  ■f^0)  as  accurately  as  may  be. 

13.  Show  that  for  large  values  of  x  the  solutions  of  ./„(x)  =  0  are  nearly  of  the 
form  IcTT  —  \  TV  -\-  \  mr  and  the  solutions  of  Kni^'^)  =  0  of  the  form  kir  +  f  tt  +  ^rnr. 

14.  Sketch  the  graphs  of  y  =  JQ{.t.)  and  y  —  '/^(x)  by  using  the  series  of  ascend- 
ijig  powers  for  small  values  and  the  asymptotic  expressions  fur  large  values  of  x. 

15.  From  Jy(x)  =  -    I     cos  (x  cos 0)r?0  show    /      e-"-'./,//)x)  Jx  =  —  ^ 

16.  Show    I      e-"'t7o(x)cZx  converges  uniformly  when  (/  s  0. 

Jo 

17.  I^valuate  the  following  integrals  :  (a)    (     JJh.i-)(lr  =  b~'^, 

^  ^  Jo 

sin  asJ^{hx)  ^  =  -  or  sin- 1  -  as  d  >  ^  >  0  ov  h  >  a.  >  0, 

0  X        2  6 

r'  1  00 

(7)    I      sin  axJ^{hx)  dx  =  -  or  0  as  a-  >  /y-  or  h-  >  a-, 

COS  ax  Jq  {bx)  dx  =  — or  0  as  b-  >  a-  or  a-  >  b-. 

0  ^  V  -  ((- 

18.  If  w  =  V^Jniax).  show  ':^  +  fr,'^  _  "'-^)u  =  0.    If  r  .=  V7rJ„(bx), 

dx-      \  X-    / 

\r-  -  u  -T=  (//-  -  «■-)  f\.h{ax)J„{bx)dx. 
L   i:/x  f?xJo  Jo 

19.  With  the  aid  of  Ex.  18  establish  the  relations: 

(a)  bJJa)J„^i{l,)  -  aJ„(b)J„^i{a)  =  Or  -  a-)  f  xJ„{ax)J„(bx)dx, 

Jo 

(13)  aJ^{a)  =  (i-  f  xJfXax)dx=    f  x.JJx)dx. 

(y)  .Tr.{«)-Tn+ii<n  +  o  [■Tnia).r„^^{«)  -  ./,>0  J« -i(«)]  =  2«  J  •'•  [•/„(ax)]2dx. 
2    r "   i^'mxtdt  ,-  ,  ,       2    r*   cnnxtdt 


CHAPTER   XV 
THE  CALCULUS  OF  VARIATIONS 
155.  The  treatment  of  the  simplest  case.    The  integral 


/  = 


Fi'-;  y,y')<-^-'- 


^{:r,  II,  d.r,  (]>/), 


a) 


where  <&  is  hoiuogemHms  of  the  first  degree  in  '/.'•  and  r///,  may  be  evalu- 
ated along  any  curve  C  Ix'tween  tlie  limits  .1  and  Ji  by  reduction  to  ;!!i 
ordinary  integral.    For  if  C  is  given  by  y  =/'(•''), 

/  =      r    Fix,  y,  y')dx  =   f  '  F(.r,  f(.r),  f'(x))d.r - 
and  if  C  is  given  by  ,/•  =  (^(O.  //  =  i/'fO, 

/  =        I       <^(.r,  I/,  <l.r.  ill/)  =    I      (P{(f>.  if/.  4>\  il/')(Jt. 

The  ordinary  line  integral  (§  122)  is  merely  the  special  case  in  which 
(J)  =  Pd.r  -f  Qdy  and  F  =^  P  +  Qy'.  In  gentn'al  the  value  of  /  will  depend 
on  the  path  C  of  integration  ;  fJu-  proldi'in  nf  tJie  cjtloi] ha  nf  nirinfinns, 
Is  to  find  tlmt  jKifli  irjildi  n-lU  nuilcc^  I  a  iDii.r'tmii  m  or  iiihi'iiiunn  ri'Idflra 
to  neigldioriufi  j»itJis. 

If  a  second  ])ath  (\  be  _//  =  /"(.'•)  +  ''/(■'■)>  where  rji-'-)  is  a  small  (pian- 
tity  Avhicli  vanishes  at  :i\^  and  x^,  a  whole  family  of  ]»aths  is  given  by 

!/  =/(■'■)  +  '^v (■''),  -  1  =  't  S  1,  y](.'\)  =  7; (./'J  =  0, 

and  the  value  of  the  inteural 


I(a)  = 


Fi.r.f-i-  ny.f  +  ar]')dx, 


(!') 


taken  along  tin'  different  [latlis  of  the  family.  1)(>-      q    -f.^ — ' J — ^ 


comes  a  function  of  a;  in  ])articular  /(O)  and  7(1) 

are  th«^  values  along  C  and  (' ^.  I'nder  a])])ropriate  assumptions  as  to 
the  continuity  of  /-'and  its  partial  derivatives  F,'.  FJ.  F'„.,  the  function 
I (<i)  will  be  continuous  and  have  a  continuous  derivative  whi(/h  may 
f)e  found  by  differentiating  under  the  sign  ( ^J  11<.»)  :   then 

^'(")  =    I        [^^■^/■'■.-  f+  '^V-f  +  "^'»  +  r]'F'„Ax.f+  orj.f  +  nrj'qdx. 

400 


6AM''    GAHU/'.-lA      «-».. 


CALCULUS  OF  VARIATIONS  401 

If  the  curve  C  is  to  give  /(a)  a  inaxiinum  or  minimum  value  for  all 
the  curves  of  this  family,  it  is  necessary  that 

^  \^^)  =f  \vK(/^  >/,  I/')  +  ■nK'i:-;  u,  y')]  '^■'-  =  o ;  (2) 

and  if  C  is  to  make  /  a  maximum  or  minimum  relative  to  all  neighboring 
curves,  it  is  necessary  that  (2)  shall  hold  for  any  function  r){x)  which  is 
small.  It  is  more  usual  and  more  suggestive  to  write  rjQr)  =  8'/,  and  to 
say  that  8//  is  the  variation  of  y  in  passing  from  the  curve  C  or  y  =  f{^r) 
to  the  neighl)oring  curve  C'  or  y  = /'(,r)  +  ^O^')-    From  the  relations 

//'  =f\^%  >l'  =/'(■'■)  +  -n'Cr),  8y'  =  riU)  =  ~py, 

connecting  the  slope  of  C  with  the  slo})e  of  C\,  it  is  seen  that  the  variation 
of  the  (In-iratire  is  the  dcrlvatlre  of  the  variation.  In  ditferential  nota- 
tion this  is  dSy  =  Sdy,  where  it  should  be  noted  that  the  sign  8  a})plies 
to  changes  which  occur  on  passing  from  one  curve  C  to  another  curve  C\, 
and  the  sign  d  applies  to  changes  taking  place  along  a  })articular  curve. 
\\'ith  these  notations  the  condition  (2j  becomes 


X 


(f;8//  +  F;.8y'j  d.r  =         SFd,-  =  0,  (3) 


where  SFis  computed  from  F,  8y,  Sy'  by  the  same  rule  as  the  differential 
dF  is  computed  from  F  and  the  differentials  of  the  variables  which  it 
contains.  The  condition  (3)  is  not  sufficient  to  distinguish  between  a 
maximum  and  a  minimum  or  to  insure  the  existenc^e  of  either;  neither 
is  the  condition  ;/'{.'■)  =  0  in  elementary  calculus  sufficient  to  answer 
these  questions  relative  to  a  function  f/ (.'■);  in  l)oth  cases  additional  con- 
ditions are  required  (§  9).  It  slundd  be  remembered,  however,  that 
these  additional  conditions  were  seldom  actually  applied  in  discussing 
maxima  and  minima  of  y(.'')  in  practical  i)roblems,  l)ecause  in  such  (;ases 
the  distinction  between  the  two  was  usually  obvious ;  so  in  this  case 
the  discussion  of  sufficient  conditions  will  be  omitted  altogether,  as  in 
»;§  '"ii^  and  (31.  and  (3)  alone  will  be  a])plied. 

An  integration  by  })arts  will  convert  (3)  into  a  differ(Mitial  ecpiation 
of  the  second  order.    In  fact 


'F'8>/'d.r  =  I     '  F;,.^8>/d.r  = 
I  '    dx 


F;,hy 


'S//-f  F>/.r 

dx     " 


Hence         f  ' ( i-^S//  +  F'^^lf) dx  =  f  '  (f;  -  ^  F'ASydx  =  0, 


(3') 


402  INTEGRAL  CALCULUS 

sinoe  the  assumption  that  B>/  =  rj(x)  vanishes  at  a-^  and  x^  causes  the 
integrated  term  [F,^,8//]  to  drop  out.    Then 

d      ,        cF        c-F  c-F     ,       c-F    „       ^ 

"       dx     ^        cy       cxCy'       cycy'  "^        cy'-"^  ^  ^ 

For  it  must  be  remembered  that  the  function  Sy  =  ■>;(,«)  is  any  function 

that  is  small,  and  if  i^^ r^  F'^,  in  (3')  did  not  vanish  at  every  point 

of  the  interval  x^  ^  .r  ^  x^,  the  arbitrary  function  hy  could  be  chosen 
to  agree  with  it  in  sign,  so  that  the  integral  of  the  product  would  neces- 
sarily be  positive  instead  of  zero  as  the  condition  demands. 

156.  The  method  of  rendering  an  integral  (1)  a  minimum  or  7naxi)7ium. 
is  therefore  to  set  vp  the  differential  equation  (4)  of  the  second  order 
and  solre  it.  The  solution  will  contain  two  arbitrary  constants  of  inte- 
gration which  may  be  so  determined  that  one  particular  solution  shall 
pass  through  the  points  A  and  B,  which  are  the  initial  and  final  points 
of  the  path  C  of  integration.  In  this  way  a  path  C  which  connects  A 
and  B  and  which  satisfies  (4)  is  found  ;  under  ordinary  conditions  the  in- 
tegral will  then  be  either  a  maximum  or  minimum.   An  example  follows. 

Let  it  be  reciuired  to  render  /  =  /       -  Vl  +  i/"-dx  a  maximum  or  minimum. 

•^x„  y 

cF  1     /- cF      y'         1 


F(x,  ?/,  y')  =  -  VI  +  ?/-,         —  ,= ^  V 1  +  y  -, 


//                           cy           y-  cy'       2/  Vl  +  y"^ 

Hence Vl  +  y"-  +  ' - if }f'  =  0     or     yy" -{■  7/"  -\- \  =  d 

y-  y-  Vl  +  y'-       2/(1  + //'-)2 

is  the  desired  equation  (4).    It  is  exact  and  the  intetiration  is  immediate. 

{yify  +1  =  0,         yy'  +  X  =  Cj.         //-  +  {X  -  ^j)-  =  r,. 

The  curves  are  circles  witli  thi'ir  centers  ow  tlie  .r-axis.  From  this  fact  it  is  easy 
by  a  freometrical  construction  to  determine  tlic  curve  wliicli  passes  tliroush  two 
fjiven  jioints  vl  (,/v„  ?/,,)  and  B{x^.  y^);  tlie  analytical  dctcnuinatinn  is  not  difticult. 
The  two  points  ^1  and  H  nnist  lie  on  the  same  side  of  the  ,f-axis  or  the  intetrral  I 
will  not  converEre  and  the  problem  will  have  no  meaning.  The  question  of  whether 
a  maximum  or  a  minimum  has  been  determined  may  be  settled  by  taking  a  curve 
Cj  whicli  lies  under  the  circular  arc  from  ,1  to  Ji  and  yet  has  the  same  length. 
The  integrand  is  of  tlie  form  ds/y  and  the  integral  along  r.  is  greater  than  along 
the  circle  C  if  y  is  positive,  but  less  if  //  is  negative.  It  therefore  appears  that  the 
integral  is  rendered  a  minimum  if  A  and  7i  are  above  the  axis,  but  a  maximum  if 
they  are  below. 

F'or  ?na?iy  2'rotile/iis  it  is  rimrc  fonrenii'nt  not  to  ))}(i];p  the  rlmicp  of  x 
or  y  as  indt'pinidott  i-oriotih'  in  the  frst  pZ/irc,  tnit  to  operate  symmetri- 
eally  vitli  }>oth  mrinhlfs  tipon  tlie  second  form  of  (X).  Suppose  that  the 
integral  of  the  variation  of  $  be  set  equal  to  zero,  as  in  (3). 


("ALCULUS   OF   VAKIATIOXS  403 

Let  the  rules  Sdx  =  (/&'•  and  8'///  =  ^/S//  be  applied  and  let  the  terms 
which  contain  d8x  and  d8i/  be  integrated  l)y  parts  as  before. 

I    8a>  =  I    [( $;  -  (/$:,,)  a/-  +  ($;  -  r/4),;;)8;/]  +  r$,;,a''  +  ^[/y^;/]  ^  =  o. 

As  J  and  B  are  fixed  points,  the  integrated  term  disap}>ears.  As  the 
variations  Sr  and  8//  may  be  arbitrary,  reasoning  as  above  gives 

^:  -  (^^:i.  =  0,       $;  -  r/$;,^,  =  0.  (4') 

If.  these  two  eipiations  can  be  shown  to  be  essentially  identical  and  to 
reduce  to  the  condition  (4)  previously  obtained,  the  justification  of  the 
second  method  will  V)e  complete  and  either  of  (4')  may  be  used  to  deter- 
mine the  solution  of  the  problem. 

Now  the  identity  4>(.r,  y.  dx.  dy)  =  F{x.  y.  dy/dx)dx  gives,  on  differentiation, 

<£'  —  F' dx  *'  —  F' dx  4>',    —  F'  *',    —  —  F'  -^  -J-  F 

by  the  ordinary  rules  for  partial  derivatives.    Substitution  in  each  of  (4')  gives 

<J>'  _  d^'    =  F'dx  -  dF'  =  i F'  -  —  F'\  dx  =  0. 

-'  '•"         -  -'        ^.    ■'      dx    -' I 

^'x  -  dK.  =  i"'A'  -  f'(^-  ^;''//')  =  K'^'^  -dF+  F'^Ah/  +  y'dF'y, 
=  F'Jx  -  F^dx  -  F'ydy  -  F;^,dy'  +  F'^^ly'  +  y'dF'^, 

=  -  Kdy  +  y'dF;,  =  -  (^F;  -  £  F;}j  dy  =  0. 

Hence  each  of  (4')  reduces  to  the  oriuinal  ciniditinn  (4).  as  was  to  be  proved. 

Suppose  this  nietli(jd  lie  applied  to    I    —  =    I    —  .  Then 

J      V         J  V 


r<U^   r.dx^  +  dy^^rVdx5dx  +  dyMy_d.l 
J       y        J  y  J    I  yds  y-       J 

=^f\d'^5x+(d^-+'^8;\, 

where  the  transformation  has  been  integration  by  parts,  including  the  discarding 
of  the  integrated  term  which  vani.shes  at  the  limits.    The  two  eqtiations  are 

d =  0.         d^^  A^ =  0  ;     and     =  — 

ydH  yds      y-  yds      c^ 

is  the  obvious  first  integral  nf  the  first.  The  integration  may  then  be  completed  to 
find  tlie  circles  as  before.  The  integration  of  the  .second  equation  wduUI  not  be  -so 
simple.  In  some  instances  t//e  advantage  of  the  choice  of  one  of  the  tico  equations 
offered  by  this  method  of  direH  operation  is  marked. 


404  I^sTEGRAL   CALCULUS 

EXERCISES 

1.  The  shortest  distance.   Treat  /  (1  +  y"-)-dx  for  a  minimum. 

2.  Treat  |  V(ir'^  +  r-d(p'^  for  a  minimum  in  polar  coordinates. 

3.  The  brachistochrone.  If  a  particle  falls  alon^^  any  curve  from  A  to  7J,  the 
velocity  acquired  at  a  distance  h  below  ^-1  is  y  =  V2(jli  regardless  of  the  path  fol- 
lowed.   Hence  the  time  spent  in  passing  from  ^1  to  B  is  T  ~  j  ds/v.    Tlie  path  of 

quickest  descent  from  A  to  B  is  called  the  brachistochrone.    Show  that  the  curve 
is  a  cycloid.    Take  the  origin  at  ^1. 

4.  The  mininuim  surface  of  revolution  is  found  by  revolving  a  catenary. 

5.  The  curve  of  constant  density  which  joins  two  points  of  the  plane  and  has  a 
minimum  moment  of  inertia  with  respect  to  the  origin  is  c^r'^  =  see  (3  (p  +  r„).  Nt)te 
that  the  two  points  must  subtend  an  angle  of  less  than  00^  at  the  origin. 

6.  Upon  the  sphere  the  mininmm  line  is  the  great  circle  (polar  coordinates). 

7.  Upon  the  circular  cylinder  the  mininmm  line  is  the  helix. 

8.  Find  tlie  mininmm  line  on  the  cone  of  revolution. 


9.  Minimize  the  integral    |      -to|"  I    +     n'-x-    t?i. 
J    l2       \dt/         2         J 


Y 

\ 

jy^ 

\B 

-^ 

■It) 

r^ 

O 

X 

157.  Variable  limits  and  constrained  minima.  This  second  luothod 
of  operation  has  also  the  advantage  that  it  suogests  the  solution  of  the 
probhiDi  of  making  an  Integral  bctirccn  rarlaldi'  aid -point  a  a  ina.rlin  ii  m 
or  vtlnunu/n.  Tims  su})i)Ose  that  the  curve  C  Avliich 
shall  join  some  point  .1  of  one  curve  F^  to  sonte 
point  B  of  another  curve  T^,  and  which  shall  make 
a  given  integral  a  minimum  or  maximum,  is  desired. 
In  the  first  place  C  must  satisfy  the  condition  (4) 
or  (4')  for  fixed  end-])oints  because  C  Avill  not  give  O 
a  maximum  or  minimum  value  as  compared  with 
all  other  curvt'S  unless  it  does  as  com})ared  merely  vvitli  all  othiM'  curves 
which  join  its  end-points.  There  must,  however,  be  additioiud  condi- 
tions which  shall  serve  to  determine  the  points  A  and  I)  whieh  C  con- 
nects.   These  conditions  are  precisely  that  the  Integrated  terms, 

Wuh'  -f-  *:,„S.v1 ''  =  0,  for  A  and  for  B,  (5) 

which  vanish  identically  Avlicn  tlu^  end-points  are  iixod.  slioJl  ranlsl/  at 
each  jiolnt  A  or  B  jirovided  hx  and  8//  are  intcr])rfled  as  diiforoutials 
aloni;-  the  curves  U  and  F, . 


CALCULUS  OF  VARIATIONS  405 


-^  ,      .      ,  ^    r  'IS        r  V(/J-  +  air  ,     ,  ,      . 

ior  fXiunple,  lu  the  case  or    j    —  =  (    treated  above,  the  integrated 

'^    y      J  y 

terms,  which  were  discarded,  and  the  resulting  conditions  are 

V±r5^      diibyl ''  ^  (hSx  +  dydijl  ^^  ^  (hdx  +  d>j5i/l    ^  ^ 

L  ijds        yds  J  A '  yds         J  '  yds         J  a 

Here  di  and  dy  are  differentials  along  the  circle  C  and  Sx  and  5y  are  to  be  inter- 
preted as  differentials  along  the  curves  Tq  and  r,^  which  respectively  pass  through 
A  and  B.  The  conditions  therefore  show  that  the  tangents  to  C  and  Tq  at  A  are 
perpendicular,  and  similarly  for  C  and  r^  at  B.  In  other  words  the  curve  which 
renders  tlie  integral  a  minimum  and  has  its  extremities  on  two  fixed  curves  is  the 
circle  which  has  its  center  on  the  jr-axis  and  cuts  both  the  curves  orthogonally. 

To  prove  the  rule  for  finding  the  conditions  at  the  end  points  it  will  be  suffi- 
cient to  prove  it  fur  one  variable  point.    Let  the  equations 

6':x  =  0(O,  y  =  ^{t),  C\:x  =  <p{t)  +  ^{t),  y  =  ^p  (t)  +  7j{t), 

nf,)  =  v(t,)  =  0,         f(g  =  «,         r,{h)=h;         Sx  =  ^{t),         Sy  =  7j{t), 

determine  C  and  C\  with  the  connnou  initial  point  .1  and  different  terminal  points 
B  and  B'  upon  T^.    As  parametric  e(iuations  of  F^,  take 

xr=x    +  al  (s),         y  =  y    +  bin  (.s) ;         --  =  al'{s),         ~  =  bm'{s), 

OS  5s 

wlicre  .s  rei^resents  the  arc  along  Fj  measured  fri)ni  B,  and  the  functions  l{s)  and 
}n  (.s)  vary  from  0  at  B  to  1  at  ]V.    Next  form  tlie  familx' 

x=^<p  (t)  +  I  (s)  f  (0,  //  =  ^  (0  -f-  m  (s)  7,  {t).         x'  =  0'  +  ^r,         y'  =  V  +  ^n7,\ 

which  all  pass  through  A  f(ir  t  =  t^^  and  which  for  t  =  t^  describe  the  curve  Fj. 
Consider 

g  (s)  :=  f  %  {x  +  I  (.s)  j-,  //  +  //( (.x)  -n.  x'  +  /j-',  y'  +  ruTj')  dt,  (6) 

which  is  the  integral  taken  fr(jm  A  to  Fj  along  the  curves  of  the  faniih',  where 
J^'  y,  •''-'■,  y'  '11'*-'  '-'11  the  curve  C  corresponding  to  s  =  0.    Differentiate.    Then 

where  the  accents  mean  differentiation  with  regard  to  ,s-  when  upon  rj.  Z.  or  ;/;.  l)ut 
with  regard  U>  t  when  on  x  or  y.  and  partial  differentiation  when  on  <l>.  and  where 
the  argument  of  <i>  is  as  in  (0).  Now  if  y  (s)  has  a  maximum  or  niininunn  when 
,s  =  0,  then 

y\0)  =  J ''  [/'(O)  r*;(,r.  y.  X'.  y')  +  m'(0)  tj*;  +  I'm  f '*;,,  -f  ?n'(0)  t?'*;.]  dt  =  0  : 

The  change  is  made  as  usual  by  integration  by  parts.    Now  as 

^(x.  y.  x'.  y')dt  =<i'{x.  y,  dx,  dy),     so     <J>^7f  =  *',..         *',.,  =  *,'/,.,  etc. 


406  INTEGRAL  CALCULUS 

Hence  the  parentheses  nnder  the  integral  sign,  when  multii)lied  by  d(,  reduce  tc 
(4')  and  vanish  ;  they  coukl  be  seen  to  vanish  also  for  the  reason  that  f  and  7;  are 
arbitrary  functions  of  t  except  at  i  =  i^  and  t  =  t-^,  and  the  integrated  term  is  a 
constant.   There  remains  the  integrated  term  which  must  vanish, 

V{0)  UK)  <.  +  m'{0)  V  (t,)  ^\y  =  [^  *;.  +  ^  *;  J'  =  [*rf,  5x  +  ¥,^  hy^  =  0. 

The  condition  therefore  reduces  to  its  appropriate  half  of  (5),  provided  that,  in 
interpreting  it,  the  quantities  5x  and  hy  be  regarded  not  as  a  =  f (f^)  and  &  =  -riii^ 
but  as  the  differentials  along  r^  at  B. 

158.  In  many  cases  one  integral  is  to  be  made  a  maximum  or  minimum 
subject  to  the  condition  tliat  another  integral  shall  have  a  fixed  value, 


1=  j      F{x,  y,  y')dx  ™',  J^  I      G  {x,  y,  y')dx  =  const. 


(7) 


For  instance  a  curve  of  given  length  might  run  from  A  to  B,  and  the 
form  of  the  curve  which  would  make  the  area  under  the  curve  a  maxi- 
mum or  minimum  might  be  desired ;  to  make  the  area  a  maximum  or 
minimum  without  the  restriction  of  constant  length  of  arc  would  b? 
useless,  because  by  taking  a  curve  which  dropped  sharply  from  .1,  in- 
closed a  large  area  below  the  ic-axis,  and  rose  sharply  to  B  the  area 
could  be  made  as  small  as  desired.  Again  the  curve  in  which  a  chain 
would  hang  might  be  required.  The  length  of  the  chain  being  given, 
the  form  of  the  curve  is  that  which  will  make  the  potential  energy  a 
minimum,  that  is,  will  bring  the  center  of  gravity  lowest.  The  pro)> 
lems  in  constrained  maxima  and  minima  are  called  Isoperliiictrle  pi-)b- 
lems  because  it  is  so  frecpiently  tlie  perimeter  or  length  of  the  curve 
which  is  given  as  constant. 

If  the  method  of  determining  constrained  maxima  and  minima 
by  means  of  undetermined  multipliers  be  recalled  (§§58,  61),  it  will 
appear  that  the  solution  of  the  isoperimetric  problem  might  reasonably 
be  sought  by  rendering  the  integral 

/  +  A./  =  r  ' [F(^x,  y,  y')  +  XG  (x,  y,  y ')]  dx  (8) 

a  maximum  or  minimum.  The  solution  of  this  problem  would  contain 
three  constants,  namely,  A  and  two  constants  c^,  e,^  of  integration.  The 
(constants  c^,  <\^  could  be  determined  so  that  the  curve  should  pass  througli 
A  and  Ji  and  the  value  of  X  would  still  remain  to  be  determined  in  such 
a  manner  that  the  integral  J  should  have  the  desired  value.  This  is 
the  method  of  solution. 


CALCULUS  OF  VAEIATIOXS  407 

To  justify  the  method  in  tlie  case  of  fixed  end-points,  which  is  the  only  casa 
that  will  be  considered,  the  procedure  is  like  that  of  §  155.  Let  G  be  given  bj 
y  =/(x)  ;  consider 

y  =f{x)  +  at]  (x)  +  /3f  (x),  7;o  =  77j  =  fy  =  s'l  =  0, 

a  two-parametered  family  of  curves  near  to  C   Then 

(J  (a,  /3)  =  J ^'F(x,  2/  +  ott;  +  iSf,  ?/  +  ^7;'+  /Sf ')  iZx,         g  (0,  0)  =  Z 

■'0 

h{a,  /3)  =  f''G{x,  y  +  cx-n  +  iSf,  V'  +  ^^V  +  iSf ')  dx  =  J  =  const. 

would  be  two  functions  of  the  two  variables  a  and  jS.  The  conditions  for  the  mini- 
nmni  or  maximum  of  (j  {a,  j3)  at  (0,  0)  subject  to  the  condition  that  h  (a,  jS)  =  const, 
are  required.    Hence 

(/^{o,  0)  +  x/C(0,  0)  =  0,      r/;(0,  0)  +  xa;j(0,  o)  =  o, 

or  f  ■'  ',7  (F,;  +  X  G;)  +  7,'(^;'  +  ^  Gy^)  dx  =  0, 

"'0 

By  integration  by  parts  either  of  these  equations  gives 

{F+\G%-^{F+\G):,  =  0;  (9) 

the  rule  is  justified,  and  will  be  applied  to  an  example. 

Required  the  curve  which,  when  revolved  about  an  axis,  will  generate  a  given 
volume  of  revolution  bounded  by  the  least  surface.    Tlie  integrals  are 

J  =  2  TT  /      yds,  min.,         J  =  it  j     y-c?.f,  const. 
^■'0  "  •^■'■(1 

Make  I     '(//cZ.s  +  \y-dx)  min.     or      |     ^5  {yds  +  \y-dx)  =  0. 

£ '' S (yds  +  XyhLc)  =: ^ ;'■'  Isyds  +  y  ^^^'^-^ +^'^^'^'^1'  +  ^  Xy5//cZx  +  X^'^ScZxl  =  0 

=  T'"^'  Fsx (-  \d  (y-)  -  d  '^\  +  5y  ids  -  d^  +  2  \ydx\~\. 

Hence  \d  {y")  +  d  ^'^'^  =  0     or     ds  -  d  —  +  2  \ydx  =  0. 

ds  ds 

The  second  method  of  computation  has  been  used  and  the  vanishing  integrated 
terms  have  been  discarded.    The  first  equation  is  simplest  to  integrate. 

^    o  1  X  (f  1  —  2/")  dy 

X2/-  +  y  - — =  =  r^X,         ±  —     ^  ^      -^  '    -^        =  dx. 

Vl  +  y'-  Vy-  -  X-  (c^  -  y-y- 

The  variables  ai'e  separated,  but  the  integration  cannot  be  executed  in  terms  of 
elementary  functions.    If,  however,  one  of  the  end-points  is  on  the  x-axis,  the 


408  INTEGRAL  CALCULUS 

values  Xy,  0,  ?/q  or  Xj,  0,  y[  must  satisfy  the  equation  and,  as  no  term  of  the  equa- 
tion can  become  infinite,  c^  must  vanisli.    The  integration  may  then  be  performed. 

•^  =  dx,         1  —  X~y-  =  X2  (x  —  c,)2     or     (x  -  (•„)    +  2/^  = 


In  tliis  special  case  the  curve  is  a  circle.  The  constants  Cj  and  X  may  be  deter- 
mined from  the  other  point  (x^,  y^)  through  which  the  curve  passes  and  from  the 
value  of  J  =  V  ;  the  equations  will  also  determine  the  abscissa  Xq  of  the  point  on 
the  axis.  It  is  simpler  to  suppose  x,,  =  0  and  leave  Xj  to  be  determined.  With  this 
procedure  the  equations  are 

A"  A"  IT         A"         o 

3  ,  o  -       *'  y     A  ^i'  +  ^i' 

or  xf  +  o  //f/, =  0,         p.,  = , 

TT  "  2x, 


and  x^-iT-l  [(3  u  -1-  V'.»  v-  +  Tr-yj^js-  -|-  (s  v  —  Vu y-  -|-  7r-(/f )']. 

EXERCISES 

1.  Show  that  {a)  the  minimum  line  from  one  curve  to  another  in  the  plane  is 
their  conunon  normal  ;  (/3)  if  the  ends  of  the  catenary  which  generates  the  mini- 
nuim  surface  of  revolution  are  constrained  to  lie  on  two  curves,  the  catenary  shall 
be  perpendicular  to  the  curves  ;  (7)  the  brachistochrone  from  a  fixed  point  to  a 
curve  is  the  cycloid  which  cuts  the  curve  orthogonally. 

2.  Generalize  to  show  that  if  the  end-points  of  tlie  curve  which  makes  any  inte- 
gral of  the  form  /  i<"{x,  y)ds  a  maximum  or  a  mininnim  are  variable  upon  two 
curves,  the  solution  shall  cut  the  curves  orthogonally. 

3.  Show  that  if  the  integrand  4>  (x,  ?/,  dx,  di/,  Xj)  depends  on  the  limit  Xj,  the 
condition  for  the  limit  B  becomes    *'/^.5x  -|-  i''/„Sy  +  dx  j    %'^.      =  0. 

4.  Show  that  the  cycloid  which  is  the  brachistochrone  from  a  point  A,  con- 
strained to  lie  on  one  curve  Fq,  to  another  curve  Y^  must  leave  F,,  at  the  point  A 
where  the  tangent  to  Fy  is  parallel  to  the  tangent  to  F^  at  the  point  of  arrival. 

5.  Prove  that  the  curve  of  given  length  which  generates  the  minimum  surface 
of  revolution  is  still  the  catenary. 

6.  If  the  area  under  a  curve  of  given  length  is  to  be  a  maximum  or  mininuim, 
the  curve  nmst  be  a  circular  arc  connecting  the  two  jpoints. 

7.  In  polar  coordinates  the  sectorial  area  bounded  by  a  curve  of  given  length  is 
a  maxinmm  or  minimum  when  the  curve  is  a  circle. 

8.  A  curve  of  given  length  generates  a  maxinuuu  or  minimum  volume  of 
revdiutiiiu.    Tile  elastic  curve 

ii  =  ^+-^:^  =  -A    or    ax  =  -^£^M^. 


CALCULU8  OF  VARIATIONS  409 

9.  A  chain  lies  in  a  central  field  of  force  of  which  the  potential  per  unit  mass  is 
V(r).    If  the  constant  density  of  the  chain  is  p,  show  that  the  form  of  the  curve  is 

dr 


+  \yh-"-  -  ly- 


10.  Discuss  the  reciprocity  of  I  and  J,  that  is,  the  cjuestions  of  making  I  a  maxi- 
mum or  minimum  when  J  is  fixed,  and  of  making  /  a  minimum  or  maxinuun  when 
I  is  fixed. 

11.  A  solid  of  revolution  of  given  mass  and  uniform  density  exerts  a  maxinunn 
attraction  on  a  point  at  its  axis.  Arts.  2\{x-  +  y-)^  +  x  =  0,  if  the  point  is  at  the 
origin. 

159.  Some  generalizations.    Supijose  that  an  integral 

/  =C  F(.r,  y,  y\  z,  z\  •  ■  •) dx  =J  ^ (.r,  dx,  y,  dy,  z,  dz,  •  •  •)       (10) 

(of  whicli  the  integrand  contains  two  or  more  dependent  variables 
//,  z,  ■  ■  ■  a,nd  their  derivatives  y\  z',  ■  ■  •  Avith  respect  to  the  independent 
variable  .r,  or  in  the  synimetrical  form  contains  three  or  more  variables 
and  their  differentials)  were  to  be  made  a  maxinmm  or  minimum.  In 
case  there  is  only  one  additional  variable,  the  problem  still  has  a  geo- 
metric interpretation,  namely,  to  find 

a  curve  in  space,  Avhich  will  make  the  value  of  the  integral  greater  or 
less  than  all  neighboring  curves.  A  slight  modification  of  the  previous 
reasoning  will  show  that  necessar}-  conditions  are 

F'  -  -^  F,:,  =  0     and     Fl  —  ~  K,  =  0 
■"       dx     ■'  -       dx     ^  (11) 

or       *;.  -  ^/$;,,,  =  0,      $;  —  d^',,,  =  o,      <i>;  —  d^[;,,  =  o, 

where  of  the  last  three  conditions  only  two  are  independent.  Each  of 
(11)  is  a  differential  equation  of  the  second  order,  and  the  solution  of 
the  two  simultaneous  equations  will  be  a  family  of  curves  in  space 
dependent  on  four  arbitrary  constants  of  integration  which  may  be  so 
determined  tliat  one  curve  of  the  family  shall  pass  through  the  end- 
points  ^l  and  B. 

Instead  of  following  the  previous  method  to  establish  these  facts,  an 
older  and  perha])s  less  accuiTite  luethod  will  be  used.  Let  the  varied 
values  of  //,  -,  //',  ,-.'',  be  denoted  by 

y  +  Sy,     z  +  8z,     y'  +  By',     z' +  6z',     Sy' =  (Sy)',     Bz' =  (Bz)'. 


410  INTEGRAL  CALCULUS 

The  difference  between  the  integral  along  the  two  curves  is 

M=  C  \f(x,  y  +  hj,  y/'  +  hj\  z  +  Iz,  z'  +  hz')  -  F{x,  y,  7j\  z,  z')yx 
=  C  \Fdx  =  C  '(F^Sy  +  F;,8y'  +  FlSz  +  f;8«')  dx  +  •  •  • , 

Jx^  Jx^ 

where  F  has  been  expanded  by  Taylor's  Formula*  for  the  four  variables 
y,  y',  z,  z'  which  are  varied,  and  ''  +  ..."  refers  to  the  remainder  or  the 
subsequent  terms  in  the  development  Avhich  contain  the  higher  powers 
of  8y,  By',  Bz,  8z'. 

For  sufficiently  small  values  of  the  variations  the  terms  of  higher 
order  may  be  neglected.  Then  if  A/  is  to  be  either  positive  or  nega- 
tive for  all  small  variations,  the  terms  of  the  first  order  which  change 
in  sign  when  the  signs  of  the  variations  are  reversed  must  vanish  and 
the  condition  becomes 


r  \F;,By  +  F;^.8y'  +  FlBz  +  F^.Bz')  dx  =  C  'SFdx  =  0.  (12 


Integrate  by  parts  and  discard  the  integrated  terms.    Then 


IK 


f;,-£f-\.,  +  (f:~£k,]b. 


0.  (13) 


*  In  tlie  simpler  case  of  §  155  this  formal  development  would  run  as 

and  with  the  expansion  A/=  51  -\ d~I  +  —  5^1 -\-  •  •  •  it  would  appear  that 

8r=f''\F;8>f+F'^,Su')<J?;        m  =   T' {Fy^jhu'^  +  2  f;;.5;/S^/'  +  F'^^^W^dx, 
■'o  -'o 

53/=  r\F'„''8!i^  +  3  f','^.:,^mi-8]i'  +  :^  F,;;;,5//5//'2  +  Fiy^w^yix.  •  •  • . 

The  terms  5/,  5-/.  5-'/.  ■  •  ■  are  called  the./?r.>-■^  ftecoml,  third,  ■  ■  ■  I'ariatioyix  of  the  inte<rral 
I  in  the  case  of  fixed  limits.  The  condition  fur  a  maximum  or  minimum  then  hecomes 
S/=  0,  just  as  ilf/  ~  0  is  the  condition  in  the  case  of  g  (.>•).  In  the  case  of  variable  limits 
there  are  some  modifications  appropriate  to  the  limits.  This  method  of  procedure  suu- 
gests  the  reasiiii  rliat  S.'",  5.'/  are  frequently  to  be  treated  exactly  as  differentials.  It  also 
suggests  that  5-/  >  0  and  5-/  <  0  would  be  criteria  for  distinguishing  between  maxima 
and  minima.  The  same  results  can  be  had  by  differentiating  (V)  repeatedly  under  the 
sign  and  exi)an(ling  /  (a)  into  .series;  in  fact,  5/=  /'(O),  52/=  F'(0),  ■  ■  ■  .    No  emphasis 

has  been  laid  in  tiie  text  on  the  suggestive  relations  dl  =  I  SF'Ix  for  fixed  limits  or 
5/=  /  54>  for  variable  limits  (variable  in  x,  y,  but  not  in  t)  because  only  the  most  ele- 
mentary results  were  desired,  and  the  treatment  given  has  some  advantages  as  to 
modernitv. 


CALCULUS  OF  VARIATIONS  411 

As  8//  and  8z  are  arbitrary,  either  may  in  particular  be  taken  equal  to 
0  while  the  other  is  assigned  the  same  sign  as  its  coefficient  in  the 
parenthesis  ;  and  hence  the  integral  would  not  vanish  unless  that  coeffi- 
cient vanished.  Hence  the  conditions  (11)  are  derived,  and  it  is  seen 
that  there  would  be  precisely  similar  conditions,  one  for  each  variable 
J/,  z,  ■  ■  ■,  no  matter  how  many  variables  might  occur  in  the  integrand. 

Without  going  at  all  into  the  matter  of  proof  it  will  be  stated  as  a 
fact  that  the  condition  for  the  maximum  or  minimum  of 

I  <I>  {x,  dx,  y,  dy,  z,  dz,  . . .)     is       /  8$  =  0, 

which  may  be  transformed  into  the  set  of  differential  equations 

of  which  any  one  may  be  discarded  as  dependent  on  the  rest ;  and 
^'aM  +  ^dM  +  *rf.8'^  H =  0;         at  ^  and  at  B, 

where  the  variations  are  to  be  interpreted  as  differentials  along  the  loci 
upon  which  A  and  B  are  constrained  to  lie. 

It  frequently  happens  that  the  variables  in  the  integrand  of  an  inte- 
gral Avhich  is  to  be  made  a  maximum  or  minimum  are  connected  by  an 
equation.    For  instance 


/ 


*(,r,  dx,  y,  dy,  z,  dz)  min.,  S{x,  y,  z)  =  0.  (14) 

It  is  possible  to  eliminate  one  of  the  variables  and  its  differential  by 
means  of  ^^  =  0  and  proceed  as  before ;  but  it  is  usually  better  to 
introduce  an  undetermined  multiplier  (§§58,  61).      From 

s{x,y,z)  =  o    follows    .s;;&/- +  .s';8y  +  5;s,v  =  0 

if  the  variations  be  treated  as  differentials.    Hence  if 


/ 


[($;  -  f/$;,,)  hx  +  (<^;  -  d^',^)  Sy  +  (4.;  -  ./$;,,)  s,^]  =  o, 


[(*;  -  d^',,  +  \s:)  8x  -f  (%  -  d^:,,,  +  XS;)  8y 

+  (^:-d^:,^-^-xs:)8z]  =  o 

no  matter  what  the  value  of  A.  Let  the  value  of  X  be  so  cliosen  as  to 
annul  the  coefficient  of  8,^.  Then  as  the  two  remaining  variations  are 
independent,  the  same  reasoning  as  above  will  cause  the  coefficients  of 
Sx  and  8//  to  vanish  and 

^:  -  "'*,/,,.  +  A.v;  =  0,  %  -  d^'„j  +  x.%  =  0,  $:  -  ./$,%  +  A.s;  =  o  (is) 


412 


INTEGRAL   CALCULUS 


■will  hold.    These  equations,  taken  Avith  S  =  0,  will  determine  y  and  z 
as  functions  of  ./•  and  also  incidentally  will  iix  A. 

Consider  the  problem  of  determining  the  shortest  lines  upon  a  surface 
S(x,  1/,  z)  =  0.    These  lines  are  called    the  geodesies.    Then 

(l.rh.r  -\-  (hjhil  +  flzhz 


Chls  =  0  =  - 


ds 


-f 


,l'$^,-  +  ,l'^S,,  +  d'$B. 


/(''  S  +  ''')  '■■■  +  (''  I'  +  ^"')^*  +  (''  I  +  ^*=) 


ds 

8y  + 


ds 


ds 
8z'. 


,  (16) 


0, 


dx 


fZ  —  +  an; 

ds 


<^ll 


+  an:  = 


s;  =  r/f +  an:  =  o, 

ds 


and 


4' 

fls 

s' 


In  the  last  set  of  equations  A  has  l)een  eliminated  and  the  equations, 
taken  with  .S'  =  0,  may  be  regarded  as  tlie  differential  equations  of  the 
rjpodesies.  The  denominators  are  proportional  to  the  direction  cosines 
of  the  normal  to  the  surface,  and  the  numerators  are  the  components  of 
the  differential  of  the  unit  tangent  to  the  curve  and  are  therefore  pro- 
portional to  the  direction  cosines  of  the  normal  to  the  curve  in  its  oscu- 
lating plane.  Hence  it  aitpeai'S  that  tlie  osculatintj  iilane  of  a,  fjeodesie 
curve  contnins  tJie  normal  tn  tlie  s\irface. 

Tlu'  iiitci^rated  terms  ri.rS/  +  f?//5//  +  <\zhz  =  0  show  that  the  least  pendesic  wliich 

connects  two  curves  on  the  svn-face  will  cut  both  curves  orthoi^onally.    These  terms 

will  also  .suffice  to  prove  a  luunber  of  interesting  theorems  which  establish  an  analogy 

between  geodesies  on  a  surface  and  straight  lines  in  a  plane.    For  instance  :  The 

locus  of  points  whose  geodesic  distance  from  a  fixed  point  is  constant  (a  geodesic 

circle)  cuts  the  geodesic  lines  orthogonal!}'.    To  see  this  write 

^7'  pr  pV  ^r  \r 

(     fZ,s  =  const..      A       (Z.s  =  0,      5  1     (?.s  1=  0,       (     5fU  -  0  -  dx5z -\- dySy -\- dz5z\    . 
Jo  Jo  Jo  Jo  I 

The  integral  in  (10)  drops  out  because  taken  along  a  geodesic.  This  final  equality 
establishes  the  perpendicularity  of  the  lines.  The  fact  also  follows  from  the  .state- 
ment that  the  geodesic  circle  and  its  center  can  be  regarded  as  two  curves  between 
which  the  shortest  distance  is  tlie  distance  measured  along  any  of  the  geodesic 
radii,  and  that  the  radii  must  therefore  be  perpendicidar  to  the  curve. 

160.  The  most  fundanuuital  and  important  single  theorem  of  mathe- 
matical physics  is  Hamilton's  Principle,  which  is  expressed  by  means 
of  the  calculus  of  variations  and  affords  a  necessary  and  sufficient  con- 
dition for  studying  the  elements  of  this  sul)ject.  Let  7'  be  the  kinetic 
energy  of  any  dynamical  system.  Let  A',-,  }'■,  Z,-  be  the  forces  which 
act  at  any  point  .r,-,  //,-,  z,-  of  tlie  system,  and  let  &/■,■,  8//,-,  S^-,-  represent 
displacements  of  that  point.    Tln'ii  the  work  is 

Sir  =  V  (A;8,ri  +  y.h;  +  ^,8,-.-,v 


CALCULUS  OF   VARIATIOXS  413 

Hamilton's  Principle  states  that  the  time  Integral 

j    \ST  +  BW) dt  =  r  '[Sr  +  2^  (X8.r  +  ]%  +  Z8.?)] dt  =  0     (17) 

riinishes  for  the  actinil  motion  of  the  si/stem.  If  in  particular  there  is 
a  potential  function  V,  then  S]r=  —  81'  and 

r  \(T  -  y)dt  =  h  f  \r-  V)dt  =  0,  (17') 

and  t/ie  time  Intcgyol  of  the  dlfferenec  hrtwccn  the  hlnetle  and  potmithd 
energli't^  Is  a  r/iaxlmum  or  mlnimuni  for  the  actual  motion  of  tit e  x^/stem 
as  coni})ared  with  any  neighboring  motion. 

Suppose  that  the  position  of  a  system  can  be  expressed  by  means  of  n  independ- 
ent variables  or  coordinates  q^.  q.^.  ■  ■  ■.  q,,.    Let  the  kinetic  energy  be  expressed  as 

^=5)  2'«'lf  =/l''"-d»l  =   T{q^,  q.^.  ■  •  .,  q„,  q^.  q.^,  ■  ■  •,  q„), 

a  function  of  tlie  coordinates  and  their  derivatives  witli  respect  to  the  time.  Let 
tlie  work  done  by  displacing  the  single  coonlinate  qr  be  5  11'  =  QrSq,--  ^''>  that  the  total 
work,  in  view  of  the  independence  of  the  coordinates,  is  Q^?qi+  Q./lq.,+  •  •  •  +  Qn'lQn- 
Then 

0  =f\dT+5W)dl  =j\l-5q,  +  7;;^5r/,  +  •  •  ■  +  7:>/„  +  T'.Sq^  +  T;^5q^ 

+  •  •  •  +    ^l'')Jnn  +    QiS^i  +   q.oq.^  +  •  •  •  +   Q„dq„)dt. 

Terform  the  usual  integration  by  ])arts  and  discard  the  integrated  terms  which 
vanish  at  the  limits  t  =  ^^  and  t  =  t^.    Then 

" = X.''  [  l'^'.  + "'  -  i  '■-)  '■"  +  ('''.  +  '•'=  - ;« '■'=)  *'= 


dt. 


In  view  of  the  independence  of  the  variations  57^.  S7.,.  ■  •  •,  S^,,, 

■'^-:^-iI=Q,        l^-l^  =  Q..        ....        '-'^-''^=q„.    (18) 

dt  cq^       cq,^  dt  cq.,       cq.-,  '  dt  a},,       cq„ 

Tiiese  are  the  Laqrangian  cquatiott--^  for  the  motion  of  a  dynamical  sj'stem.*    If 
there  is  a  potential  function  I"  {q^,  ry.,.  •  •  •,  q„).  then  by  detinition 

C     -  -'Jl         c     ~-^Jl  c    -  -—         L^-iT-         -LJI-o 

d  cL       cL               d  cL       cL                         d  cL       cL 
Hence =  0. =0.     ••., =  0.      L  =  1  —   \  . 

dt  cq^       cq^  dt  cq.2       cq^  dt  cq„       cqn 

The  equations  of  motion  have  l.)een  expressed  in  terms  of  a  single  function  L.  -which 
is  the  difference  between   the  kinetic   energy    T  and   potential    function    V.    By 

*  Cniiiiiari'  L\.  10,  p.  1V2.  for  a  deduction  of  Hs)  hy  transforniation. 


41i  IXTEGEAL  CALCULUS 

comparing  the  equations  -with  (17')  it  is  seen  that  the  dynamics  of  a  sj^stem  whicli 
may  be  specified  by  n  coordinates,  and  wliich  has  a  potential  function,  may  be  stated 

as  the  problem  of  rendering  the  integral   /  Ldt  a  maximum  or  a  minimum  ;  both  the 

kinetic  energy  T  and  potential  function  V  may  contain  the  time  t  without  chang- 
ing the  results. 

For  example,  let  it  be  required  to  derive  the  equations  of  motion  of  a  lamina 
lying  in  a  plane  and  acted  upon  by  any  forces  in  the  plane.  Select 'as  coordinates 
the  ordinary  coordinates  (x,  i')  of  the  center  of  gravity  and  the  angle  ^  through 
which  the  lamina  may  turn  about  its  center  of  gravity.  The  kinetic  energy  of  tlie 
lamina  (p.  318)  will  then  be  the  sum  ^Mv"  +  ilu}-.  Now  if  the  lamina  be  moved  a 
distance  5x  to  the  right,  the  Avork  done  by  the  forces  will  be  X5x,  where  A'  de- 
notes the  sum  of  all  the  components  of  force  along  the  x-axis  no  matter  at  wliat 
points  they  act.  In  like  manner  Y5y  will  be  the  work  for  a  displacement  dy.  Sup- 
pose next  that  the  lamina  is  rotated  about  its  center  of  gravity  through  the  angle 
50;  the  actual  displacement  of  any  point  is  rdcp  where  r  is  its  distance  from  the 
center  of  gravity.  The  work  of  any  force  will  then  be  Iird(p  where  R  is  the  com- 
ponent of  the  force  perpendicular  to  the  radius  ?■ ;  but  Kr  =  4>  is  the  moment  of 
the  force  about  the  center  of  gravity.    Hence 

T=  1 3r(x2  4-  y^)  +  1 702,         5 11'  =  X5x  +  Y5y  +  <l>50 

(l^x  d-y  d-4> 

and  M —  =  A,         ^[ —  =  1 .         I —  =  *, 

dt^  dt-  dt- 

by  substitution  in  (18),  are  the  desired  equations,  where  X  and  1'  are  the  tctal 
Cdinponents  along  the  axis  and  <i>  is  the  todd  nmnient  about  the  center  of  frravity. 
A  particle  glides  withnut  friction  on  the  iiiterii:)r  of  an  inverted  cone  of  revo- 
lution ;  determine  the  motion.  Choose  the  distance  r  of  the  particle  from  the  ver- 
tex and  the  meridional  angle  0  as  the  two  coordinates.  If  /  be  the  sine  of  the 
angle  between  the  axis  of  the  cone  and  the  elements,  then  ds-  =  dr-  +  f-l-drp-  and 
f-  =  r-  ■\-  r-l-4>~.  The  pressure  of  the  cone  against  the  particle  does  no  work  ;  it  is 
normal  to  the  motion.  For  a  change  50  gravity  does  no  work;  for  a  change  8r  it 
does  work  to  the  anioinit  —  //if/ V 1  —  l-dr.    Ilt/nce 


T  =  I  m  (/'•'-  -f  /■-/-0-).          5  ir  =  —  //?;/>  1  —  /-5/-     or     V  =  mr/\'\  —  Pr 
d-r         ,o /'^'^\'"  /^j ,o  '^  I  .y,nd(p\      ^  „d(f) 


Then rl-    —      =-r/Vl  -  P.         -    r^l-  —    =  0     or     f^    ^-  =  C 

df^  \dt)  ^  dt\        dtj  dt 

The  remaining  integrations  cannot  all  be  effected  in  terms  of  elementary  functions. 
161.   Sui)post'  the  double  integral 

extended  over  a  certain  area  of  the  , ''//-plane  were  to  be  made  a  maxi- 
mum or  minimum  by  a  surface  z  =  z(.r,  y),  Avliich  shall  ]iass  through  a 
given  curve  upon  the  cylinder  wliich  stands  ujton  tlic  bounding  curve 
of  the  area.    This  prol)lem  is  analogous  to  the  problem  of  §  l.j.")  with 


CALCULUS  OF  VARIATIONS  415 

fixed  limits  ;  the  procedure  for  finding  the  partial  differential  equation 
which  z  shall  satisfy  is  also  analogous.    Set 

jfsF>/.n/>/  =  fuF'.hz  +  F;,hp  +  F'^h'i)dxdij  =  0. 

C?i"  ??>•' 

Write  hp  —  -—-  '  87  =  -~  and  integrate  by  parts. 

The  limits  A  and  B  for  which  the  first  term  is  taken  are  points  upon 
the  bounding  contour  of  the  area,  and  8.~  =  0  for  A  and  B  by  virtue  of  the 
assumption  that  the  surface  is  to  })ass  through  a  fixed  curve  above 
that  contour.  The  integration  of  the  term  in  Sy  is  similar.  Hence  the 
condition  becomes 

jjmu,,,  =//(k  - 1:  f  -  ;^,  f )  a.-.«.  =  o        (20, 

l^--fl^-^i^'=0,  (2O0 

cz       dx  c^t       dtj  C'l  ^      ^ 

by  the  familiar  reasoning.    The  total  differentiations  give 

F'  —  F"  —  F"  —  F" p  —  F"'t  —  F"  r  —  2  F" s  —  F"f  =  0. 

The  stcck  illustration  introduced  at  this  point  is  the  minimum  surface, 
that  is.  the  surface  Avhich  spans  a  given  contour  Avith  tlie  least  area  and 
which  is  pliysically  rci)resented  by  a  soap  him.  The  real  iise,  however, 
of  the  theory  is  in  connection  with  Hamilton's  Princijde.  To  study  the 
motion  of  a  chain  hung  u}i  and  allowed  to  vibrate,  ov  of  a  })iano  Avire 
stretched  between  two  ]ioints.  compute  the  kinetic  and  })otential  energies 
and  a})ply  Hamilton's  Principle.  Is  tin;  motion  of  a  vibrating  elastic 
body  to  be  investigated  ?  A]»])ly  Hamilton's  Principle.  And  so  in 
electrodynamics.  In  fact,  with  the  very  foundations  of  mechanics  some- 
times in  doubt  owing  to  modern  ideas  on  electricity,  the  one  refuge  of 
many  theorists  is  Hamilton's  Principle.  Two  problems  Avill  be  Avorked 
in  detail  to  exliiljit  the  method. 

Let  a  uniform  chain  of  density  p  and  leniith  I  be  suspended  by  one  extremity 
and  caused  to  execute  small  oscillations  in  a  vertical  j^lane.  At  any  time  the  shape 
of  the  curve  is  y  ■=  y{x).  and  y  =  // (w.  t)  will  he  taken  to  represent  the  shape  of  the 
curve  at  all  times.  Let  y'  =  cy/c.r  and  y  =  cy/ct.  As  tlie  oscillations  are  small, 
the  chain  will  rise  only  sliizhtly  and  the  main  part  of  the  kinetic  eneri,'y  Avill  be  in 
the  whippinu-  motion  from  side  to  side  ;  the  assumption  dx  =  ds  may  be  made  and 
the  kinetic  energy  may  be  taken  as 


-x>©- 


416  INTECJPvAL  CALCULUS 

The  potential  energy  is  a  little  harder  to  compute,  for  it  is  necessary  to  obtain  the 
slight  rise  in  the  center  of  gravity  due  to  the  bending  of  the  chain.  Let  X  be  the 
shortened  length.    The  position  of  the  center  of  gravity  is 

x= — =  -X—       I        -X x\y~ax. 

f    (1+  ly'-')dx         x+  /    ly'-dx 

Here  ds  =  Vl  +  y''^dx  has  been  expanded  and  terms  higlier  than  y'"  have  been 
omitted. 

l  =  \+  \      -  y'-dx,         ~l-x=\      -  (X  -  X)  y'Hx,         V=lpgl~l-x]. 

Ji\     2  2  \Ji)     2  \2  I 

Then  f'\T-  V)  dt  =  C''  C'\\p  {^■^\  dx  -  ^-^  pg{l  -  x)  (^)"1  dxdt,  (21) 

provided  X  l)e  now  replaced  in  V  by  I  wliicli  differs  but  slightly  from  it. 

Hamilton's  Principle  states  that  (21)  must  be  a  maximum  or  mininuun  and  the 
integrand  is  of  precisely  the  form  (19)  except  f<ir  a  change  of  notation.    Hence 

d  V  ,,        .cyl       d(   cy\       .  Ic^ij      ,,        .c-y      cy 

_         _pj/(l_3.)_    _       L,^_   \  =0     or     ^        =(i_x)~,-— . 
dx\_  cx_\      dt\    ctf  get-  ex-      ex 

The  change  of  variable  I  —  x  =  u-,  which  brings  the  origin  to  the  end  of  tlie  chain 
and  reverses  the  direction  of  the  axis,  gives  the  differential  equation 

e-i/       ley      i  c'-y  d-P      1  dP  ,   4)i2 

-^  H = or \ 1 P  =  0     if     y  =  P  (u)  cos  nt. 

cu^      u  ell      g  et-  dii^       u  du         g 

As  the  equation  is  a  pai'tial  differential  ecpiation  the  usual  device  of  writing  the 
dependent  variable  as  the  pmduct  of  two  functions  and  trying  for  a  special  type 
of  solution  has  l)een  used  (§  1U4).  The  etiuation  in  P  is  a  Eessel  equation  (§  107) 
nf  whicli  one  sohuion  P{u)  =  A  J ^^  {2  ng~ '-^  u)  is  tinite  at  the  origin  u  =  0,  while  the 
other  is  inlinite  and  must  be  discarded  as  not  representing  possible  motions.    Thus 

y{x,  t)  =  vl,7„(2  ng~2u)cni>nt,     with     y  {I,  t)  =  AJ^^{2rig~  ^l^  =  0 

as  the  condition  that  tiie  chain  shall  l)e  tied  at  the  original  origin,  is  a  possible 
UKidc  of  motion  for  the  chain  and  consists  of  whipping  back  and  forth  in  the  peri- 
odic time  27r/?i.  The  condition  J,,(2ng~-i-)  =  0  limits  n  to  one  of  an  infinite  set 
of  values  obtained  from  the  roots  (jf  ./,,. 

Let  there  be  found  the  equations  for  the  motion  of  a  medium  in  which 

v  =  ]  '^fffU'  +  'J'  +  ^'')  ''■^•''//'-^^ 

are  the  kinetic  and  i)otential  energies,  where  A  and  B  are  constants  and 

4  tt/  = .  I  irg.  -.      ,         4  tvIi  = ~ 

cy      cz  '        Iz      Ix  ex      cy 


CALCULUS   OF  VAKIATIO^'S  417 

are  relations  connecting/,  (/,  /(  with  the  displacenients  ^,  17,  f  along  the  axes  of  x,  ?/,  z. 
Then 

ffffs[U  ik-'  +  V-  +  h  -  hB  if-  +  g~  +  h-)-]  dxdydzdt  =  0  (22) 

is  the  expression  of  Hamilton's  Principle.  These  integrals  are  more  general  than 
(19),  for  there  are  three  dependent  variables  ^,  77.  f  and  four  independent  variables 
X,  y,  z,  t  of  which  they  are  functions.  It  is  therefore  necessary  to  apply  the  method 
of  variations  directly. 

After  taking  the  variations  an  integration  by  parts  will  be  applied  to  the  varia- 
tion of  each  derivative  and  the  integrated  terms  will  be  discarded. 

jyjjs  I  A  (e  +  ^-  +  t-)  dxdydzdt  =  ffff-^  m  +  ^5^  +  tst)  dxdydzdt 

=  -  ffff^^  (^^^  +  V^V  +  fSr)  dxdydzdt. 
ffffs  i  B{f'  +  r/-  +  h") dxdydzdt  =   CCCC B{f5f+  r/8(j  +  !i8h) dxdydzdt 

=-iX0'£fi-S)«-(S-S"-g-i) '*]-'"- 

After  substitution  in  (22)  the  coefficients  of  5f.  5?;,  5j"  may  be  severally  equated  to 
zero  because  5^,  5?;,  5j' are  each  arbitrary.    Hence  the  equations 

dt^  \cy      czl  (t-  \cz      cxj  It"  \cx      cyl 

With  the  proper  determination  of  A  and  7>  and  the  proper  interpretation  of  ^,  r;,  f', 
/,  (J,  h,  these  are  the  equations  of  electromagnet  ism  for  the  free  ether. 

EXERCISES 

1.  Show  that  the  straight  line  is  the  shortest  line  in  spacer  and  that  the  shortest 
di.stance  between  two  curves  or  surfaces  will  be  normal  to  both. 

2.  If  at  each  point  of  a  curve  on  a  surface  a  geodesic  be  erected  perpendicular 
to  the  curve,  the  locus  of  its  extremity  is  perpendicular  to  the  geodesic. 

3.  "With  any  two  points  f)f  a  surface  as  foci  construct  a  geodesic  ellipse  l)y  tak- 
ing the  distances  FP  +  F'P  =  2  a  along  the  geodesies.  Show  that  the  tangent  to 
the  ellipse  is  ecjually  inclined  to  the  two  geodesic  focal  radii. 

P  r 

4.  Extend  Ex.  2.  p.  408.   to  space.    If    /     F{x.  y.  z)dn  =  cornet.,  show  tliat  the 

Jo  ' 

locus  of  P  is  a  surface  normal  to  the  railii,  provided  the  radii  be  curves  which 

make  the  integral  a  maxinmm  or  minimum. 

5.  Obtain  the  polar  equations  for  the  motion  of  a  particle  in  a  plane. 

6.  Find  the  polar  equations  for  the  motion  of  a  particle  in  space. 

7.  A  particle  glides  down  a  helicoid  {z  =  k<p  in  cylindrical  coordinates).  Find 
the  equations  of  motion  in  (r,  (p),  (r,  z).  or  {z,  <p),  and  carry  the  integration  as  far 
as  possible  toward  expressing  the  position  as  a  function  of  the  time. 


418  INTEGRAL  CALCULUS 

8.  If  z  —  ax^  +  hy-  +  •  •  • ,  witli  a  >  0,  6  >  0,  is  the  Maclaurin  expansion  of  a 
surface  tangent  to  the  plane  z  =  0  at  (0,  0),  find  and  solve  the  equations  for  the 
motion  of  a  particle  gliding  about  on  the  surface  and  remaining  near  tlie  origin. 

9.  Show  that  r(l  +  (p-)  +  i(l  +  p'-)  —  2_p(/s  =  0  is  the  partial  differential  equa- 
tion of  a  minimum  surface  ;  test  the  lielicoid. 

10.  If  p  and  6'  are  the  densitj'  and  tension  in  a  uniform  piano  wire,  show  that 
the  approximate  expressions  for  the  kinetic  and  jjotential  energies  are 

2  Jo  ^\ci/  2  Jo      \cx} 

Obtain  the  differential  equation  of  the  motion  and  try  for  solutions  ?/  =  P(x)  cos  nL 

11.  If  ^,  1),  fare  the  displacements  in  a  uniform  elastic  medium,  and 

cz  cy  cz  \dij      czl  \cz      cx/  \cx      cyf 

are  six  conil)inations  of  the  nine  possible  first  partial  derivatives,  it  is  assumed  that 

V  =  I  j  I  Fdxdydz,  where  Fis  a  homogeneous  (juadratic  function  of  a,  b,  c^f,  g,  h, 

with  constant  coefficients.    Establish  the  equations  of  the  motion  of  the  medium. 

c-2|       c^F        c^'F        c-F  c^-n       r~F       c"-F        c^-F 

P  —\  — 1 1 '         P  —  ~ h h  -     -  . 

ct-      dxda      cych      czcg  ct^       cxch      cijcb      czcf 

€"-•<:       c-^F        d"F        c^F 

P^,  = 1 1 

ci-      cxcg      cycf      czlc 

12.  Establish  the  conditions  (11)  by  the  method  of  the  text  in  §  1.5-5. 

13.  By  the  method  of  §  159  and  footnote  establish  the  conditions  at  the  end 
points  for  a  minimum  of    |  F{x,  y,  y')dx  in  terms  of  F  instead  of  <l>. 

14.  Prove  Stokes's  Formula  /  =    C¥-dx  =  ffVxF.dS  of  p.  .34.5  by  the  calculus 

of  variations  along  the  following  lines  :  First  compute  the  variation  of  I  on  pass- 
ing from  one  closed  curve  to  a  neighboring  (larger)  one. 

dl  =  d  f  F.dr  =  f  {5F.dr  -  rfF.Sr)  +  f  (Z(F.5r)  =  f  (VxF).(5rx<'7r), 

Jo  Jo  Jo  ^G  ' 

where  the  integral  of  '/ (F.5r)  vanishes.  Second  interpret  the  last  expression  as 
the  integral  of  VxF-dS  over  tlie  ring  formed  by  one  position  of  the  closed  cur\e 
and  a  neighlxiring  ])ositi<)n.  Finally  sum  up  the  variations  5/  which  thus  arise  on 
passing  througli  a  succ('ssi(}n  of  closed  curves  expanding  from  a  point  to  tinai  coin- 
cidence with  the  given  closed  curve. 

15.  In    case    the    integi'and   contains  //"  sliow    by    successive    integrations  liy 
parts  that 

5  I       F{x,  y,  y'.  y")dx  =     }  'a,  +  1  "a;'  -     --  co      +  }  _  +  -    ccdx, 

•^.T„  L  dx       J 11     J.,„   \  (//         ax-  / 

w  =  5y. 


.r       if'^ 

(F 

(F 

where 

Y  =        , 

y 

Y" 

d!/ 

?,'/ ' 

~  W' 

PART  lY.    THEORY   OF   FUNCTIONS 

CHAPTER    XVI 
INFINITE    SERIES 
162.  Convergence  or  divergence  of  series.*    Let  a  series 

=  %  +  '/j  +  ",  +  •••  +  '^,-l  +  «„  +  •  •  • ,  (1) 


X 


If  = 


the  terms  of  Avhicli  are  constant  but  infinite  in  number,  be  given.  Let  tlie 
sum  of  tlie  first  ?i  terms  of  tlie  series  be  written 

'^  =  "o  +  "i  +".,+  ••■  +  ''„-i  =  X  "•  (2) 

Then  S^,  .v.,  S.^,  ■  ■  ■ ,  ,<„  S,,^„---  ° 

form  a  definite  suite  of  numl^ers  -wliieli  Jtun/  approach  a  definite  limit 
lim  ^„  =  S  when  n  becomes  infinite.  In  tliis  case  tlie  series  is  said  to 
conrerge  to  the  value  S,  and  S,  Avliich  is  tlie  limit  of  the  sum  of  the  first 
ji  terms,  is  called  the  sum  of  the  series.  Or  \  vk///  not  ajjproach  a  I  unit 
when  n  becomes  infinite,  either  because  the  values  of  ^'„  become  infinite 
or  because,  though  remaining  finite,  they  oscillate  about  and  fail  to 
settle  down  and  remain  in  the  vicinity  of  a  definite  value.  In  these 
cases  the  series  is  said  to  Jicerge. 

Tlie  necessary  and  sufficient  condition  that  a  series  converge  Is  that  a 
ralue  of  n  may  he  found  so  large  that  the  numerical  value  of  ^'„  +  p  —  >'„ 
sJiall  he  less  than  any  assigned,  value  fir  every  value  <f  pj-  (See  §21, 
Theorem  3,  and  compare  p.  356.)  A  sufficient  condition  that  a  series 
diverge  is  that  the  terms  u,^  do  not  approach  the  limit  0  when  n  becomes 
intinite.  For  if  there  are  always  terms  numerically  as  great  as  some 
number  r  no  matter  how  far  one  goes  out  in  the  series,  there  must 
always  be  successive  values  of  X„  which  differ  by  as  much  as  /•  no 
matter  how  large  n,  and  hence  tlie  values  of  ,S^„  cannot  possibly  settle 
down  and  remain  in  the  vicinity  of  some  definite  limiting  value  ^. 

*It  will  be  useful  to  read  over  Chap.  II,  §§  lS-22,  and  Exercises.  It  is  also  advisable 
t<;>  compare  niatiy  of  the  results  for  infinite  series  with  the  corresponding  results  for 
iiifiuite  integrals  (Chap.  XIII). 

419 


420  THEORY  OF  FUNCTIONS 

A  series  in  which  the  terms  are  alternately  positive  and  negative  is 
called  an  altcrndilnrj  series.  An  aJ.ternatin(j  scries  iii  tcltich  the  terms 
approach  0  as  a  liriut  when  n  hecomcs  infinite,  each  term  being  less  than 
its  2y>'<i'^lscessorj  vlll  converge  and  the  dljference  between  tJiesuni  S  of  the 
series  and  the  sum  >S'„  of  the  Jirst  n  terms  Is  less  than  the  next  term  v ^^. 
This  follows  (i).  39,  Ex.  3)  from  the  fact  that|  .S;^^,  -  .S'„l  <  //„  and  ?/„  =  0. 

For  example,  consider  the  alternating  series 

1  -  X-  +  2  J-*  —  3 .r«  H +  (-  l)«)ix2"  +  •  •  •  . 

If  |x|  ^  1,  the  individual  terms  in  the  series  do  not  approach  0  as  n  becomes  infinite 
and  the  series  diverges.    If  |x]  <  1,  the  individual  terms  do  approach  0  ;  for 

1-           1         1-         '1           ,•                  1  ^ 

Inn  )i.i'-"  =   lim  =   Inn =  0. 

(,  ■=  z  K  =  x  /-  - "       ),  =  X  —  2  J-  -  "  log  X 

And  for  sufficiently  large*  values  of  7i  tiie  successive  terms  decrease  in  magnitude 
.since                                                                           .,  i 

vu;-"  <  ()(,  —  l).c-«-^     gives     >  x-     or     n  > 


1  -  X- 


Hence  the  series  is  seen  to  converge  for  any  value  of  x  numerically  less  than  unity 
and  to  diverge  fnr  all  other  values. 

The  Compakisox  Test.  If  the  terms  of  a.  series  are  all  pasltlre  (or  all 
ne;/aflre^  (Hid  each  term  l^  name/'lealli/  less  tlian  the  corresponding  term 
(fa  series  of  jinsltlve  terms  vhleh  Is  Iniown  to  converge,  the  series  con- 
verges and  tlie  dljference  S  —  .S',,  Is  less  than  the  corresp(jmhng  difference 
fnr  tlte  series  Irnowyi  to  co^iverge.     ((."f.  p.  355.)    Let 

"u  +  "l  +  "■.'  +  •  •  •  +  "„_i  +  "„  -\ 

and  ii[,  -f  v[  +  ii'..  -{-■•■  -\-  ii'„_i  +  ^'n  +  ■■■ 

be  respecti\'fl_v  the  given  series  and  the  series  known  to  converge. 
Since  tlie  terms  of  the  first  arc  less  than  those  of  the  second, 

s,,^^,  -  s„  =  //„  +  •  •  •  +  y„^-,,-^  <  ":  +  •••  +  K^,^i  =  '^U;,  -  •^:- 

Xow  as  the  second  quantity  >S',', +,,  —  >',',  can  he  made  as  small  as  desired, 
so  can  the  first  ([uaiitity  .S^„u.^,  —  >'„,  which  is  less  ;  and  the  series  must 
converuv.    The  remainders 


/',.  =  .S' 


"n   +    ".-1    H =2    "• 


^'-K  =  "■  +  "1-1  +  •■■  =2  "' 


*It  should  be  remarknl  tliat  tlio  bcliavinr  of  a  serirs  near  its  Ijeginiiing  is  of  no  oon- 
seqiu'iice  in  regard  to  its  (•(uiviTgi'iici'  or  liivergeiice  :  the  tirst  .V  terms  may  be  added 
and  considered  as  a  finite  sum  .S' v  and  the  series  may  be  written  as  .S'\-4-  )i\+  ".v^i  -r  •  ■  • : 

it  is  tlie  properties  of  Uy  +  (ly  +  i  -\ which  are  important,  that  is,  the  ultimate  behavior 

of  the  series. 


iXFIXlTE   SERIES 


421 


clearly  satisfy  the  stated  relation  7'„  <  U\^.    The  series  which  is  most 
frequently  used  for  comjjarison  with  a  given  series  is  the  geometric, 


a  +  ar  -\-  <u^  +  av^  + 


R.= 


0  <  r  <  1, 


(3; 


Avhich  is  known  to  converge  for  all  values  of  r  less  than  1. 
For  example,  consider  the  series 

1 

+ 


1  +  1  +  -  +  ^-  + 


1 


and 


1  +  1  + 


1 


+ 


2  ■  3  ■  4 
1 


2-2 


+ 


+  —  + 


+  ^1  + 


Here,  after  tlie  first  two  terms  of  the  first  and  the  first  term  of  tiie  second,  each 
term  of  the  second  is  greater  than  tlie  C(jrresponding  term  of  tlie  first.  Hence  the 
first  series  converges  and  tlie  remainder  after  the  term  1/n  1  is  less  than 

2"      2''+i  2"  1  —  -1       2"-i 

A  better  estimate  of  the  remainder  after  the  term  1/n  !  iwax  be  had  bj^  comparing 
11.  .,1  1  1 


i'„  = 


■,^+7 ,+ 

(u  +  1)  :       {n  +  2)  ! 


with 


(H  +  1)  : 


+ 


(u  +  l)!(/t  +  l)       '"       n\n 


163.  As  the  convergence  and  divergence  of  a  series  are  of  vital  im- 
portance, it  is  advisable  to  have  a  nuniher  of  tests  for  the  convergence 
or  divergence  of  a  given  series.  The  test 
by  comparison  with  a  series  known  to  con- 
verge requires  that  at  least  a  few  types  of 
convergent  series  be  known.  For  the  estal>- 
lishment  of  such  ty}ics  and  for  the  test 
of  many  series,  the  terms  of  which  are 
positive,  Cinirhy's  Interjrifl  fi'st  is  useful. 
Suppose  that  the  terins  of  tlie  series  are 
decreasing  and  that  a  function /(«)  which  decreases  can  l)e  found  such 
that  II, ^  =f(n).  Xow  if  the  terms  //„  l)e  plotted  at  unit  intervals  along 
the  «-axis,  the  value  of  the  terms  may  be  interpreted  as  the  area  of 
certain  rectangles.  The  curve  i/=zf(n)  lies  above  the  rectangles  anJ 
the  area  under  the  curve  is 

r  fin)  (In  >  >'.,  -f  ;/g  +  . . .  +  u„.  (4) 

Hence  if  the  integral  convcrgt'S  (which  in  practice  means  that  if 

I  f{n)dii  =  F{ji),     then      /     j\a)  =  -F(x)  —  F{1)  is  tinitcj, 


422  THEOrvY   OF   FUNCTIONS 

it  follows  that  the  series  inust  converge.    For  instance,  if 

be  given,  then  u„  —f(n)  =  l/n",  and  from  the  integral  test 


1      1 

f"  (hi 

^  +  7r  +  - 

•  <  1 



2p      3p 

./ 

,         71" 

provided  ])  >  1.  Hence  the  series  converges  if  //  >  1.  This  series  is 
also  very  useful  for  comparison  with  others  ;  it  diverges  if  p  ^  1 
(see  Ex.  8). 

Thk  R.\tio  Test.  If  tlte  ratio  of  tico  sticcessli-e  ti'rms  in  a  series  ofjiosi- 
tire  terms  approaches  a  limit  which  is  less  tlnni  1,  tlte  series  converges ; 
if  the  ratio  approaches  a  limit  which  is  greater  than  one  or  if  the  ratio 
becomes  uifiuite,  the  series  diverges.    That  is 

if  lim  — ^"-^^  =  y  <  1,  the  series  converges, 

if  lim  -^^-^  =  y'  >  1,  the  series  diverges. 

For  in  the  first  case,  a.s  the  ratio  approaches  a  limit  less  than  1.  it  must  be  pos- 
.sible  to  go  so  far  in  the  series  that  the  ratio  shall  be  as  near  to  7  <  1  as  desired, 
and  hence  shall  be  less  than  /•  if  r  is  an  assigned  number  between  7  and  1.    Then 

Un  +1  <  rUn  ,  Un  +  C  <  '""/i  +1  <  '•-«„  .  •  •  • 

and  j/„  +  u„  +1  +  "«  +  2  +  •  •  •  <  "«(1  +  '■  +  '•-  +  •••)  =  "« :. 

1  —  /• 

The  proof  of  the  divergence  when  »,i+i/w«  becomes  infinite  or  approaches  a  limit 
greater  than  1  consists  in  noting  that  the  individual  terms  cannot  apprnach  0.  Note 
that  if  the  limit  of  the  ratio  is  1.  no  in/urination  relative  to  the  convergence  or 
divergence  is  furnished  bj-  this  test. 

If  the  scries  of  numerical  or  absolute  values 

h'o!+i^i+;"j+---+!^.i+--- 

of  the  terms  of  a  series  wliich  contains  jxjsitive  and  negative  terms 
converges,  tlie  series  converges  and  is  said  to  concerfje  absolatelij.  For 
consider  the  two  sums 

■^'„+y.  -  \=  ".  H Viin^p-x     and     |/^,|^ h|"„  +  ;,_il. 

The  first  is  surelyriot  numerically  greater  than  the  second:  as  the 
second  can  be  made  as  small  as  desired,  so  can  the  first.  It  follows 
therefore  that  the  given  series  must  converge.    The  converse  proposition 


INFINITE   SERIES  423 

that  if  a  series  of  positive  and  negative  terms  converges,  then  the  series 
of  absolute  values  converges,  is  not  true. 

As  an  example  on  convergence  consider  the  binomial  series 

m  (m—  \)   „      in  (in  —  1)  (?/i  —  2)    „  7?i  (m  —  1)  •  •  •  {in  —  n-\-\\ 

1-2  1-2-3  1-2. ..ji  ' 

h'n  +  il       \m  —  ?l|  I"--  •  '1 

where  ' — ^-^ —  ■ \'^\i  hm 


l"ni  71  +  1  n=:c    jl<„| 

It  is  therefore  seen  that  the  limit  of  the  quotient  of  two  successive  terms  in  the 
series  of  absolute  values  is  |j|.  This  is  less  than  1  for  values  of  x  numerically  less 
than  1,  and  hence  for  such  values  the  series  converges  and  converges  absolutely. 
(That  the  series  converges  for  positive  values  of  x  less  than  1  follows  from  the  fact 
that  for  values  of  n  greater  than  m  +  1  the  series  alternates  and  the  terms  approach 
0 ;  the  proof  above  holds  equally  for  negative  values.)  For  values  of  x  numerically 
greater  than  1  the  series  does  not  converge  absolutely.  As  a  matter  of  fact  when 
\x\  >  1.  the  series  does  not  converge  at  all ;  for  as  the  ratio  of  successive  terms  ap- 
proaches a  limit  greater  than  unity,  the  individual  terms  cannot  approach  0.  For 
the  values  x  =  ±  1  the  test  fails  to  give  information.  The  conclusions  are  there- 
fore that  for  values  of  |j|<l  the  binomial  series  converges  absolutely,  for  values 
of   |x|>l  it  diverges,  and  for  |X|  =  1  the  question  remains  doubtful. 

A  word  about  series  with  comphx  terms.    Let 

^^;,  +  "i  -\■y.^ h  "«- 1  +  «„  H 

=  »o  +  ^I'x  +  ''2  H +  ^<,  _  1  +  ?<,  H 

+  H"o'  +  y'i  +  "o'  H h  '',;_  1  +  v"^  -\ ) 

be  a  series  of  complex  terms.  The  sum  to  n  terms  is  >'„  =  .S',',  +  iS'^. 
The  series  is  said  to  converge  if  S^^  approaches  a  limit  \rhen  n  becomes 
infinite.  If  the  com})lex  number  \  is  to  approach  a  limit,  both  its  real 
part  aS^j  and  the  coefficient  N,7  of  its  imaginary  part  must  approach  limits, 
and  hence  the  series  of  real  parts  and  the  series  of  imaginary  parts 
must  converge.  It  will  then  be  possible  to  take  n  so  large  that  for  any 
value  of  p  the  simultaneous  inequalities 

\^'n^„->^'n\<\^     and     |5:;^^, -.^:|<  |e, 

where  e  is  any  assigned  nundjer,  hold.    Therefore 

Hence  if  the  series  converges,  the  same  condition  holds  as  for  a  series 
of  real  terms.    Xow  conversely  the  condition 

1^+,, -."<„! <e    implies    [s;.^, -.s:,;<c,       |,s;^^^_5:j<6. 

Hence  if  the  condition  liolds,  the  two  real  series  converge  and  the  com- 
plex series  will  then  converge. 


424  THEORY  OF  FUNCTIONS 

164.  As  Cauchy"s  integral  test,  is  not  easy  to  apply  except  in  simple  cases  and 
the  ratio  test  fails  when  the  limit  of  the  ratio  is  1,  other  sharper  tests  for  conver- 
gence or  divergence  are  sometimes  needed,  as  in  the  case  of  the  binomial  series 
when  X  =  ±  I.    Let  there  be  given  two  series  of  positive  terms 

Iti,  +  Ul  +  ■■■  +  Kn  +  ■■■      and      Vi)  +  Vi  +  ■  ■  ■  +  v„  +  ■  ■  ■ 

of  which  the  first  is  to  be  tested  and  the  second  is  known  to  converge  (or  diverge). 
//  the  ratio  of  two  .swcce.s.stwe  terrns  Un  +  i/Un  ultimately  becomes  and  remains  less  {or 
greater)  than  the  ratio  u,i  +  i/u,i,  the  first  series  is  also  convergent  {or  divergent).    For  if 

U„  +  1         l'«  + 1  Un  +  2        Vii  +  2  . ,  Un        Un  + 1        W„  +  2 

<^  J  '^  ,      ' ' '  1      tneii  ^  ^  s> ' ' ' . 

"rt  y«  U„  +  1         Vn  +  l  Vn         Vn  _|.  1         Vn  +  2 

Hence  if     u„  =  pv,,,         then     h„  +  i  <  pr„  +  ,,         h„  +  o  <  pv^  +  2,         •  •  • , 

and  Un  +  w«  +  i  +  Wn  +  2  +  ••■  <  piv^  +  Vn^\  +  u„  +  2  +  •••)• 

As  the  y-series  is  known  to  converge,  the  pij-series  serves  as  a  comparison  series 
for  the  w-series  which  must  then  converge.  If  ?<„  +  ]/«„  >  u,j  +  i/i'„  and  the  i;-series 
diverges,  similar  reasoning  would  show  that  tlie  «-series  diverges. 

This  theorem  serves  to  establish  the  useful  test  due  to  Eaabe,  which  is 

if    lim  n(-  ^ 1 )  >  1,  S^  converges  ;       if   lim  n  I — 1 )  <  1.  <S'„  diverges. 

Again,  if  the  limit  is  1,  no  information  is  given.  This  test  need  never  Ije  tried 
i.-\c,ept  wlien  the  ratio  test  gives  a  limit  1  and  fails.    Tlie  proof  is  simple.    For 


and 


r-         dn  1        1       I"-    ..    • 

I =  — IS  fnnte 

J      i((logn)i  +  "^  a- (log  )t)''J 

r  ^-      dii  ,       ,         "1  '■"..,,    . 

I      =  li)gl()gn      IS  mnnite, 

«/       n  log  n         '  J 


1  1  ,1  1 

hence 1-  •  •  •  H 1-  •  •  ■     and h  ■  •  •  -1 h  ■  •  • 

2  (log  2)1  +  ^  n(log7iy+''  i^(l<>g^)  71  (log  n) 

are  respectively  convergent  and  divergent  by  Cauchy's  integral  test.    Let  these  be 
taken  as  the  i;-seri(;s  with  wliicli  to  compare  the  H-series.    Then 

J'l^  ^  "  +  2  /l'>g(H  +  l)y +  "^  A  ^  1\  /log(]  +>oY  +  ^ 

l"«  f  1  n      \       log  71       /  \  7// \       log  71       / 

r„  /,  l\]nLc(1+7() 

and  ^  —  =    1  +  -  )  -— V^J — ' 

Vn+l         \  nj  log  71 

in  the  two  respective  cases.    Next  consider  Raabe's  expression.    If  first 
Ii)ii7i( — ~ 1)>^,     then  idtimately     ni — "-  —  1)>7>1     and     — "- >  1 + 

Now        lim    ^^'    ^-')        =1     and  ultimately     (l^£il'-^^)        <  1  +  e, 

».  =  z\        l(lg)t         /  \        l0g7i        / 


IIS^FINITE  SERIES  425 

where  e  is  arbitrarily  .small.    Hence  ultimately  if  7  >  1, 

\        )(/  \      log  n      /  n         n-  n 

or  Vn/Vn  J- 1  <  ",,/Mh  + 1      Or     ii„  +  i/u„  <  v,,  +  i/y„, 

and  the  u-serie.s  converue.s.    In  like  manner,  .secondly,  if 

lira  ni~ 1  )<  1.     then  ultimately     — -  <l  +  '^,         7  <  1  ; 

and  1  +  -i  <  ( 1  +  -  )     '^^      — '     or     — —  <  — ^     or     -^^^  >  -^^^ . 

71        \  )(/  log  71  Wh+1         r„fi  It,,  i'rt 

Hence  a.s  the  r-series  now  diverges,  the  i/-series  nuist  diverge. 

Suppo.se  this  test  applied  to  the  binomial  series  for  x  =  —  1.    'J'lien 

"«  n  +  1  ,.  /n  +  1        A  ?7t  +  1 

=  —      - ,  lim  71 1 1 )  =  ''™    =  "^  +  1- 

7(„+i       n  —  in  n  =  j:     \n  —  m        /       »  =  x  ;/( 

n 

It  follows  that  the  .series  will  converge  if  m  >  0,  but  diverge  if  m  <  0.  If  .r  =  +  1, 
the  binomial  series  becomes  alternating  for  n  >  vt  +  1.  If  the  .series  of  absolute 
values  be  considered,  the  ratio  of  .sttccessive  terms  |  «„/«„  + 1 1  Ls  still  {n  +  l)/{n  —  in) 
and  the  binomial  series  converges  absolutely  if  m  >  0  ;  but  when  ?7i  <  0  the  series 
of  absolute  values  diverges  and  it  remains  an  open  question  whether  the  alternat- 
ing series  diverges  or  converges.    C(.)nsider  therefore  the  alternating  series 

1  +  ,71  +  f^^i^^-'^)  ,  "1  ("i -_1) (m-^)  ,  ,  m{m--[).--{m-n  +  -i)  ^^ 

1-2  1.2.3  l"----^  1.2. ..71  ' 

This  will  converge  if  the  limit  of  u,,  is  0,  but  otherwi.se  it  will  diverge.  Now  if 
ni  =  —  1,  the  successive  terms  are  multiplied  b\'  a  factor  ]?«  —  ?(  +  1  \/n  s  1  and 
they  cannot  approach  0.  When  —  1  <  ?7t  <  0,  let  1  +  ?/i  =  ff.  a  fraction.  Then  tiie 
7tth  term  in  the  series  is 

and  -  log  1  u„  ]  =:  _  log  (1  -  (9)  -  log  /l  -  ^^ log  (l  -  ^)  • 

Kach  successive  factor  diminishes  the  term  but  diminishes  it  by  so  little  that  it  may 
not  approach  0.    The  logarithm  of  the  term  is  a  series.    Now  apply  (Jauchy's  test. 

f  '  -  log  (l  -  -\  dn  =  l-ri  log  (l  -^\  +  d  log  {n  -  ^)1 '  =  co. 

The  series  of  logarithms  therefore  diverges  and  lim|w„|=e-^-  =  0.  Hence  the 
terms  approach  0  as  a  limit.  The  final  results  are  therefore  that  when  x  =  —  1  the 
binomial  series  converges  if  m  >  0  but  diverges  if  in  <  0  ;  and  when  ./;  =  +  !  it  con- 
verges (absolutely)  if  m  >  0,  diverges  if  7n  <  —  1,  and  converges  (nut  ab.solutely)  if 
—  1  <  ?/i,  <  0. 


420  THEOllY   OF  FUNCTIONS 

EXERCISES 

1.  State  the  number  of  terms  which  must  be  taken  in  these  alternating  series  to 
obtain  the  sum  accurate  to  three  decimals.  If  the  number  is  not  greater  than  8, 
compute  the  value  of  the  series  to  three  decimals,  carrying  four  figures  in  the  work  : 

I  1  1  1  1  1  1  1 

.      s     ,  1  1  1  /^^  1  1  1 

(^>^-2  +  3-4  +  ---'  (^^k^-i^  +  h:;^^-""-' 

(0   1-^  +  :?-, -^  +  ---,  (f)  e-i-2e--2  +  3e-3_4e-*  +  .... 

y-     o-      I  - 

2.  Find  the  values  of  x  for  which  these  alternating  series  converge  or  diverge: 

1,1.  .   ^    ,       a^"       a^"*       a-s 

{a)   1  _  X-  +  -x^  -  -./•"  +  .  .  . ,  (/i)   1  _++... , 

2.  o  z  ;       4  :       u  . 

3.3  j-h  J.7  ,  ^  J^         .<■■'         .'■' 

(7)  X  --  +  --.,+••• ,  (5)  X  --+-.---  +  ••• , 

X-      x*       x6  ,  ^    ^  23  X-'       2''x''       2'x" 

6     !--  +  ----  +  •••,  (f    2x-  -^  +  — ^  +  ---, 

1'^       2^'       3>'  0  o  ( 

II  1  1  1  2  2-  23 

(^) + +  .  .  (^) + +  .  . . . 

X      X  +  1       .'J  +  2      X  +  3  X      X  +  1       X  +  2      X  +  3 

3.  Show  that  these  series  converge  and  estimate  the  error  after  ?t  terms  : 

,       .       1        1        1  ,  ox   1       1  •  2       1  ■  :^  •  3 

2-       6^      4*  3       o  ■  o      3  •  o  ■  / 

,11  1  1  ,   ,   /IV-      /I  •:^\-      /I  •2-3\- 

From  the  estimate  of  error  state  how  many  terms  are  required  to  compute  the 
series  accurate  to  two  decimals  and  make  the  computation,  carrying  three  figures. 
Test  for  convergence  or  divergence  : 

(e)  sni  1  +  sin     +  sm- +  •  •  •,  (j-)  siii- 1  +  sin-     +siii--  +  ---, 

2  3'  23 

(77)  tan-i  1  +  tun-'  -  +  tan-'  -  +  ...,      (())  tan  1  H =  tan  -  f  -  =  tan  -  +  ..., 

2  3  ^'2         -       \  3         •' 

1  +  1       2  + A  2       3+\3  2-—  1-       o- —  2-       4- —  o- 

,,1        2       2.3       2.3.4  ,1        \'^>       ^  .3       v4 

X         X-  X3  X''  X  X-  X-^  X* 

4.  Apply  Cauchy's  integral  to  determine  the  convergence  or  divergence  : 

,       ,       I0-2      loir  3      log  4  ,  ^  ^  1  1  1 

2/'  3/'  4/'  ^^^  2(iou-2)/'       3(loi.^3)/'       4(log4)/> 


IXFIXITE   SERIES  427 

CO                            ,                                                                    rr-. 
(•y)l+X^    ,  (5)1+^   , 

■^  ?i  log  n  log  log  )i  ^  n  log  n  (log  log  n)i' 

2  3  4 

(e)  cot-i  1  +  cot-i  2-1 ,  (f)  1  + h 1 +  •  •  •  • 

5.  Apply  the  ratio  test  to  detennine  convergence  or  divergence  : 

^12        3        4  ,.v      22        23        2* 

(''^)2  +  i^  +  23  +  2^  +  ---'  ^^^     ^i  +  .3To  +  iro  +  ---' 

2  !       3  '       4  '       5 '  22       33       44 

^^^2^  +  3^+4^+5^  +  ---'  ^^)2l  +  ^  +  4l  +  ---' 

910  ?!10  410 

(e)   Ex.  3(<r).  (/3),  (7).  (5) ;  Ex.  4(ar),  (f),         (f)  --  +  —  +  -^-^^-^  +  •  •  -, 

('')i  +  ^  +  ^;  +  o:  +  -'-'  (^>i  +  2.  +  4.  +  ---' 

, ,  x^      3.3      2-4  ,  ,    1      bx'     h-x^ 

2        3        4  a      a"       a^ 

6.  Where  the  ratio  test  fails,  discuss  the  above  exercises  by  any  method. 

7.  Prove  that  if  a  scries  of  decreasing  positive  terms  converges,  lim  ?z«„  =  0. 

8.  Formulate  the  Cauchy  integral  test  for  divergence  and  clieck  the  statement 
on  page  422.    The  test  has  been  used  in  the  text  and  in  Ex.  4.    Prove  the  test. 

9.  Show  that  if  the  ratio  test  indicates  the  divergence  of  tlie  series  of  absolute 
values,  the  series  diverges  no  matter  what  the  distribution  of  signs  may  be. 

10.  Show  that  if  \  u„  approaches  a  limit  less  than  1,  the  series  (of  positive 
terms)  converges;  but  if  a  u,,  approaches  a  limit  greater  than  1.  it  diverges. 

11.  If  the  terms  of  a  convergent  series  u^  +  u^  -f-  u.^  +  •  •  •  of  positive  terms  be 
multiplied  respectively  by  a  set  of  positive  numbers  a^^,  n^.  o.,.  ■  ■  ■  all  of  which  are 
less  than  some  number  G,  the  resulting  series  (i^^Uq  +  a^ui  +  (i.,u.,  +  ■  ■  ■  converges. 
State  the  corresponding  theorem  for  divergent  series.  What  if  the  given  series  has 
terms  of  opposite  signs,  but  converges  absolutely  ? 

in     „,         ,,    ,  ,,  .      sin  X       sin2x      sin3.r       sin4x  , 

1^.  Show  that  the  series 1 — h  ■  ■  ■  converges  abso- 

1-2  22      ^      32  42 

lutcly  for  any  value  of  ,r.  and  that  the  series  1  +  x  sin  0  +  x^  sin  20  +  x''  sin  S$  +  ■  ■  ■ 

converges  absolutely  for  any  x  immerically  less  than  1,  no  matter  what  d  may  lie. 

13.  If  ftp,  cfj,  «.2,  •  •  ■  are  any  suite  of  numbers  such  that  -v/i«n|  approaches  a 
limit  less  than  or  equal  to  1,  show  that  the  series  a^  +  a^x  +  a„x^  +  ■  •  ■  converges 
absolutely  for  any  value  of  x  numerically  less  than  1.  Apply  this  to  show  that  the 
following  series  converge  absolutely  when  |x|  <  1 ; 

1  1-3  1 • 3  •  5 

(a)  1  +  -  x2  + X*  + j-6  +  .  . .       («)  1  _  2  X  +  3  x2  -  4  x3  +  •  •  • , 

^    ^  2  2 .  4  2  •  4 .  i;  ^  ' 

(7)  1  +  X  +  2px^  +  3j'x^  +  4px*  +  •••,        (5)  l-xlogl  +  x2  1og4-x3  1og9  +  •■•. 


428  THEOKY  OF  FUNCTIONS 

14.  Show  tliat  in  Ex.  10  it  will  \k'  .sufficient  for  convergence  if  ^u„  becomes 
and  remains  less  than  7  <  1  without  approaching  a  limit,  and  sufficient  for  diver- 
gence if  there  are  an  iiitiinty  of  values  for  n  such  that  -y/un  >1.  Note  a  similar 
generalization  in  Ex.  13  and  state  it. 

15.  If  a  power  series  a,,  +  (l^x  -f  n....r"  +  a.,/^  +  ■  ■  ■  converges  for  x  =  X>0,  it 
converges  ab.solutely  for  any  x  such  that  \x\  <  A',  and  the  .series 

a,/  +  I  a^X"  +  I  (i.-,x-^  +  ■  ■  ■     and     a^  +  2  (i.-,x  +  3  (ux"  +  •  •  • , 

obtained  l)y  integrating  and  differentiating  term  liy  term,  also  converge  absolutely 
for  any  yalue  of  x  such  that  |.r|  <  A'.  The  same  result,  by  the  .same  proof,  holds  if 
the  terms  (/j,,  "jA",  «.,A"-,  ■  •  •  remain  less  than  a  fixed  value  (J. 

16.  If  the  ratio  of  the  .successive  terms  in  a  .series  of  positive  terms  be  regarded 
as  a  function  of  l/ji  and  may  be  expanded  by  Maclaurin's  Formula  to  give 

"?)  1       M-  /1\"  .    .       ^   .  1       „ 
=  a  +  p — l^l^?         At  remaining  unite  as  -  =  0, 

the  .series  converges  if  tir  >  1  or  a  —  I,  ^  >  1,  but  diverges  if  a  <  1  or  a  =  1,  p^l. 
This  test  covers  mo.st  of  the  .series  of  po.sitive  terms  which  arise  in  practice.  Apply 
it  to  various  instances  in  the  text  and  previous  exercises.  Why  are  there  series  to 
which  this  test  is  inapplicable  ? 

17.  If  Pi,,  pj.  p.T.  •  ■  •  is  a  decreasing  .suite  of  positive  mimbers  approaching  a 
limit  X  and  Sq,  6\,  .s'.,.-  •  •  is  any  limited  suite  of  number.s,  that  i.s,  numbers  such 
that  ["S,,!  ^  G.  .show  that  the  .series 

{Po  -  Pi)  '"^0  +  (Pi  -  P2)  '^i  +  ^P-2  -  P.-)  ■'^■1  +  •  •  •  converges  ab.solutely, 


and 


^(P„-P«M)-S/^    Oip.,-^)- 


18.  Apply  Ex.  17  to  show  that.  p,,.  p,.  p.,.  ■  •  •  being  a  decreasing  suite,  if 

"0  +  "1  +  "■■  +  ■  ■  ■  converges.         p^^ii^^  -\-  p^ii^  +  p.,»(.,  +  ■  ■  •  will  converge  also. 

X.H.     p„(/o  +  pi»i  +  •  •  •  +  PnUn  =  p„Ni  +  pi  (N,  -  .s\)  +  ...  +  p„  (.S,  .1  _  S„) 

=  S^  Ip^,  _  pj)  +  .  .  .  +  s„  (p„  _i  -  p„)  +  p„S„+i. 

19.  Apply  Ex.18  to  prove  Ex.  l-")  after  showing  that  p,,*/,,  +  p,?(,  +•••  must 
converge  al)si)lutely  if  p,,  +  pj  +  •  •  •  (■(uiverges. 

20.  If  r/p  ((„.  (I.,.-  ■  ■.  (f„  are  ;;  jxisitixc  luimliers  less  than  1.  shriw  that 

(1  +  "1)  (1  +  ",)  •■•(!+  n„)  >  1  +  «i  +  ",  +  •■•  +  "„ 
and  (1  —  a,)  (1  —  r/.,) ...(]  —  r^,,)  >  l  _  ,/j  _  ,/ ,  _  .  .  .  _  ,i„ 

by  induction  or  any  other  method.    Then  since  1  +  ''j  <  1/(1  —  ^'j)  show  that 

>(1  +  0  ){■[  +  ,1.,)  •  ■  ■  (1  4-  "„)  >  1  +  ("j  +  "..  +  •  •  •  +  a„). 

1  -  («!  +  a.,  +  •  •  •  +  ««) 

-     ^  -  "*- -— -  >(1  -  a,)  (1  -  a.,)  ■••(!-  a„)  >  1  -  {a,  +  a,  +  ■■■  +  a„), 

1  +  (",  +  ".,  +  •  •  ■  +  ('„) 


INFINITE   SERIES  429 

\i  a^  +  0.^  +  ■  ■  ■  +  a„  <  I.    Or  if  JJ  be  the  symbol  for  a  2)roduct, 

\      1    '       ^  I  \      1    /       ^  1 

21.  Let  TT(1  +  "i)  (1  +  "n)- •  •  (1 +"„)  (1 +"«ri)- ■  •  l""  ^'1  iiitiiiite  product  and 
let  P„  be  the  product  of  the  first  n  factors.  Show  that  j  P„  +  ,,  —  P„  1  <  e  is  the  neces- 
sary ami  sufficient  condition  that  P„  approach  a  limit  when  n  becomes  infinite. 
Show  that  u„  must  approach  0  as  a  limit  if  P„  approaches  a  limit. 

22.  In  case  P„  approaches  a  limit  different  from  0.  .show  that  if  e  be  a.s.signed, 
a  value  of  n  can  be  found  .so  large  that  for  any  value  of  p 

;^±1'  -  1 1  =  "tt'  (1  +  »,)  -  1 1  <  e     or     "fr"  (1  +  ",)  =  1  +  ^.         \v\<^. 

\      I'n  I         ,"-1  I  "^1 

Conversely  show  that  if  this  relation  holds,  P„  must  approach  a  limit  other  than  0. 
The  infinite  product  is  said  to  converge  when  P„  approaches  a  limit  other  than  0  ;  in 
all  other  ca.ses  it  is  .«aid  to  diverge,  including  the  case  where  lim  P„  —  0. 

23.  By  combining  Exs.  20  and  22  .show  that  the  nece.s.sary  and  .sufficient  con- 
dition that 

P„  =  (l  +  a^)  (1  +  rt„) .  •  .  (1  -f  r,„)     and     Q,,  =  0  -  ",)  (1  -«.,)••■  (1  -  a,,) 

converge  as  n  becomes  infinite  is  that  tiie  .series  ((^  J-  (/.,  +  ■  ■  ■  +a„  +  ■  ■  ■  sliall  coi:- 
verge.  Note  that  P„  is  increasing  and  Q„  decreasing.  Sliow  that  in  ca.se  Za  diverges, 
P„  diverges  to  cc  and  Q„  to  0  (provided  ultimately  «,  <  1). 

24.  Define  ab.solute  convergence  for  infinite  products  and  .show  that  if  a  product 
converges  ab.solutely  it  converges  in  its  oritrinal  form. 

25.  Test  these  products  for  convergence,  divergence,  or  absolute  convergence: 

(7 )  fr  Fi  -  (jT^jV]  •  (5 )  (1  +  ■^) (1  +  .'■-) ( 1  +  .H) (1  +  -n  •  •  • . 

(.)(i-    '-)U ^\(i__-L\....      (nfrr(i--^)en. 

^         \         ]og2/\         (log  4)-^'^  (log  8)3/  VIA         c  +  nl       I 

26.  (iiveu  --—  or     u-  <  n  —  lou(l  -|-  ;/)  <     (/-  or    -        accf>rdini,'-  as  u  is  a  iiosj- 

1  -^  M        2  2  \  +  u 

tive  or  negative  fraction  (.M-e  K.\.  2'.».  p.  ]1).    I'rovc  that  if  S;;;,  converges,  then 

«n  +1  -h   Un  +  2  +  •  •  •   +  Un+p  -  log  (1  +  U„  +])  (1   +   U„  ,,  o)  .  .  .  (l-f  H  „  ^^,) 

=  (Sn^p-Sn)-(\<^?Pn^>,-\"?'P«) 

can  lie  made  :'.s  small  as  desired  by  taking  n  large  enough  regardless  of  p.  Ilence 
prove  that  if  In'i  converges,  ffd  +  !(„)  converges  if  "Zu,,  does,  but  diverLit's  to  x 
if  2m„  diverges  to  +  x  .  and  diverges  to  0  if  Z'-(„  diverges  to  —  x  ;  when-as  if  lit'l 
diverges  while  2«„  converges,  the  product  diverges  to  0. 


430  THEORY  OF  FUI^CTIONS 

27.  Apply  Ex.  26  to :       (a)  (l  +  2)  (^  "  l)  (^  +  i)(^  "  I, 

<«(-^)(-7l)('-7l)--     «(-i)('-?)(-f)('-?)- 

28.  Suppose  the  integrand /(x)  of  an  infinite  integral  oscillates  as  x  becomes  in- 
finite.  What  test  might  be  applicable  from  tlie  construction  of  an  alternating  series  ? 

165.  Series  of  functions.    If  the  terms  of  a  series 

S(x)  =  u^(:r)  +  u^  (.r)+  •  •  ■  +  >',X^)  +  •  •  ■  (6) 

are  functions  of  x,  the  series  deiines  a  function  Si^x)  of  x  for  every 
value  of  X  for  which  it  converges.  If  the  individual  terms  of  the  series 
are  continuous  fun(;tions  of  x  over  some  interval  a  ^  x  ^  h,  the  sum 
.S„  (x)  of  n  terms  will  of  course  be  a  continuous  function  over  that  inter\al. 
Suppose  that  the  series  converges  for  all  points  of  the  interval.  Will  it 
then  be  true  that  .S^.'"),  the  limit  of  <S'„(.t),  is  also  a  continuous  function 
over  the  interval  ?    Will  it  be  true  that  the  integral  term  by  term, 

u^(x)dx  +  I     u^(:i')dx  +  ■  ■  ■,     converges  to       I     S(x)dx2 

U  a  mJ  a 

Will  it  be  true  that  the  derivative  tei'm  by  term, 

«o(-^')  +  "iC'')  +  ■  ■  ■?     converges  to     S' (x)  ? 

There  is  no  a  jn-lorl  reason  why  any  of  these  things  shoidd  be  true  ;  for 
the  proofs  which  were  given  in  the  case  of  finite  sums  will  not  api)ly 
to  the  case  of  a  limit  of  a  sum  of  an  infinite  numlx'r  of  terms  (cf.  §  144). 

'I'hese  ijuestions  may  readily  be  tlirown  into  the  form  of  (juestions  concerning 
the  possibility  nf  iu^•erti^l;■  the  onicr  nf  two  limits  (see  §  44). 

For  integration  :  Is  I      lim  S„  (x)(Z.c  =  lim    |     .S,  (.r) (Z.c '.' 

P'or  differentiation  :  Is  --    lim.s'„(x)=  lim— N„(,r)? 

ds  « =  X  w  =  X  d.c 

For  contiimity  :  Is  lim    limtS„(j)=  lim    linijS,j(.r)? 

.T  =  x^  «  =  /:  »  =  X  a-  =  x^ 

As  derivatives  and  definite  integrals  are  themselves  defined  as  limits,  the  existence 
of  a  double  limit  is  clear.  That  all  three  of  the  questions  mttst  be  answered  in  the 
negative  unless  some  restriction  is  placed  on  the  way  in  which  fS'„(j')  converges  to 
S(x)  is  clear  from  some  examples.    Let  0  ^  x  ^  1  and 

S„(x)  =  xji-e-"'',     then      lim  iS'„(x)  =  0,     or     <S'(x)  =  0. 

n=  X 

No  matter  what  the  value  of  x,  the  limit  of  .S'„  (x)  is  0.  The  limiting  function  is 
therefore  contiiuious  in  this  case  ;  but  from  the  manner  in  which  N,,  (x)  converges 


INFINITE  SEEIES 


431 


r 

Ij^ 

o 

}4      y^ 

1      X 

to  S  (x)  it  is  apparent  that  under  suitable  conditions  tlie  limit  would  not  be  con- 
tinuous. The  area  under  the  limit  S  (x)  =  0  from  0  to  1  is  of  course  0 ;  but  the 
limit  of  the  area  under  <S„  (x)  is 

lim   I    xn^e-^dx  =  lim    e-"-^(—  nx  —  1)      =1. 

»  =  oot/0  n  =  oo  L  Jo 

The  derivative  of  the  limit  at  the  point  x  =  0  is 
of  course  0  ;  but  the  limit, 

lim     —  {xii^e-"^) 

»  =  oo  l_dX  Ja;=0 

=  lim     yi^e-'^{l  —  nx)  =  lim  ?|2  =  ao, 

»  =  ocL  Ja-=0        n  =  -x, 

of  the  derivative  is  infinite.  Hence  in  this  case  two  of  the  questions  have  negative 
answers  and  one  of  them  a  positive  answer. 

If  a  suite  of  functions  such  as  '^^(.t),  ^^(.t),  •  •  • ,  S,^(x),  ■  ■  ■  converge  to  a 
limit  S(x)  over  an  interval  a  ^  x  ^  h,  the  conception  of  a  limit  requires 
that  when  e  is  assigned  and  x^  is  assumed  it  nnist  be  possible  to  take  n 
so  large  that  |is,(-^"o)l  =  l-^X'^'o)  ~"  ^n(^'o)\  <  ^  fo^'  this  and  any  larger  n. 
The  suite  is  said  to  converge  uniform! tj  toward  its  limit,  if  this  condition 
can  be  satisfied  simultaneously  for  all  values  of  x  in  the  interval,  that  is, 
if  when  e  is  assigned  it  is  possible  to  take  n  so  large  that  |  7l„  (,x)  |  <  e 
for  every  value  of  x  in  the  interval  and  for  this  and  any  larger  n.  In 
the  above  example  the  convergence  was  not  uniform ;  the  figure  shows 
that  no  matter  how  great  n,  there  are  always  values  of  x,  between  0  and 
1  for  which  S^^  {x)  departs  by  a  large  amount  from  its  limit  0. 

Tlie  uniform,  convergence  of  a  continuous  function  S^fx)  to  its  limit  is 
sufjjcientto  insure  the  continuitij  of  the  Vuuit  >S'(.r).  To  show  that  .S'(,t)  is 
continuous  it  is  merely  necessary  to  show  that  when  e  is  assigned  it 
is  possible  to  find  a  ^x  so  small  that  \Six  +  A,r)  —  H (.r) |  <  e.  But 
I S  (.r  +  A,r)  -  5(.r)  |  =  |  .S„  {x  +  A.r)  -  S^^  (,r)  +  7?„  (,>■  +  A.r)  -  7.'„  (,/•)  |;  and 
as  by  hypothesis  i?„  converges  uniformly  to  0,  it  is  ])ossible  to  take  n 
so  large  that  |  /.'„  (.-r -|-  Aa')  |  and  [  7i„  (,t)  |  are  less  tlian  \  e  irrespective  of  x. 
Moreover,  as  S^fx')  is  continuous  it  is  possible  to  take  A.r  so  small  that 
I  .S^„  {x  -f-  A,t)  —  5'„  (.r)  I  <  ^  e  irrespective  of  x.  Hence  |  .S"  (x  +  A.r)  —  S  (,t)  [  <  e, 
and  the  theorem  is  proved.  Although  the  uniform  convergence  of  S^^  to  ^S' 
is  a  sufficient  condition  for  the  continuity  of  .S",  it  is  not  a  necessary  con- 
dition, as  the  above  example  shows. 

The  uniform  convergence  of  S,^(x)  to  its  limit  insures  thxit 


U  a 


432 


THEORY   OF   FITXOTIOXS 


For  in  the  first  plac^e  S(t)  must  be  continuous  and  therefore  integrable. 
And  in  the  second  phice  Avhen  e  is  assigned,  n  may  be  taken  so  large 
that  I  A'„  (;r)  |<  €/(b  -  a).    Hence 

S  (.r)  dx  -    I     .S;,  (.r)  <l.r   =1     7.'„  (,r)  (h'    <    /     y^-  ./,r  =  e, 

and  tlie  result  is  proved.  Similarly  if  >%(^')  Is  c(mtlnH(>Hs  and  converges 
unlformlij  to  a  Viinit  T(:r^,  then  7'(.t)  =  S' {x^.  For  by  the  above  result 
on  integrals, 


£ 


T{x)dx  =  lim    |     Sl^(x)dx  =  lim 


.s:,(.>0 -'•?»(-) 


=  S(x)-S(a). 


Hence  T(.'')  =  S'(x).  It  should  be  noted  that  tliis  proves  incidentally 
that  if  S'^(x)  is  continuous  and  converges  uniformly  to  a  limit,  then 
S(af)  actually  has  a  derivative,  namely  T(x). 

In  order  to  apply  these  results  to  a  series,  it  is  necessary  to  have  a 
frst  for  t/ie  imlformiti/  of  the  convergence  of  the  series  ;  that  is,  for  the 
uniform  convergence  of  S„(x)  to  S(x).  One  such  test  is  Weierstrass^s 
Jf-test :   The  series 

^(•^•)+^(.T)  +  ---+'/„(.r)+---  (7) 

IV lU  converge  -iuilfon)! ///  jn'ovlded.  a,  convergent  series 

M^  +  M^  +  ..-  +  M,^  +  .--  (8) 

of  positive  terms  moi/  he  found,  such  that  nltlniatehj  \i'i(p'^\  =  ^li-  The 
proof  is  immediate.    For 

1  I'n  (•'■)  I  =  I  "„  (•'■)  +    "«  +1  (•'■)    +   •  •  -I   ^    -V„   +   3/„  +,   +   ••• 

and  as  the  iU-scu'ies  converges,  its  remainder  (!an  be  made  as  small  as 
desii'ed  by  taking  n  sufficiently  large.  Hence  any  series  of  continuous 
functions  defines  a  continuous  function  and  may  be  integrated  term  by 
term  to  find  the  integral  of  that  function  provided  an  .l/-test  series  may 
be  fomid  ;  and  the  derivative  of  tliat  function  is  the  derivative  of  the 
series  term  bv  term  if  this  derivative  series  admits  an  J/-test. 


To  apply  11h_!  work  to  an  exaniplo  consider  whether  the  series 

,, ,  ,       cos  J      CCS  2  a;      cos  3  a;  cos  )*.r 

.S  (.r)  =  ---  + + \ h +  • 

1-  2-  3-  71- 


(■') 


defines  a  continiKius  fuiu'tinn  and  may  be  integrated  and  differentiated  term  by 

term  as 

sin2.r       sin3.r 

1 


X'«<'>=>" 


+ 


+    -    .7,-  -   + 


+ 


and 


d       ,   ,  sin.c       sin2,c 

(Ic        '  1  2 


(.'''■') 


IXFIXTTE   SERIES  433 

As  |cosx|^  1,  the  convergent  series  1  -\ — ;;  +  7;;  +  ---4 — r  +  *'"  "I'^y  '-"-'  taken  as 

an  3f-series  for  S{x),  Hence  S{x)  is  a  continnons  f miction  of  x  for  all  real  values 
of  X,  and  the  integral  of  S{x)  may  he  taken  as  the  limit  of  the  integral  of  Su{x), 
that  is,  as  the  integral  of  the  series  term  by  term  as  written.  C)n  the  other  hand, 
an  j\/-series  for  (7'")  cannot  bo  found,  for  the  series  1  +  |  +  -|  +  •  •  •  is  not  conver- 
gent. It  therefore  appears  that  S' (x)  may  not  be  identical  with  the  term-by-term 
derivative  of  <S  (x) ;  it  does  not  follow  that  it  will  not  be,  —  merely  that  it  may  not  be. 

166.  Of  series  with  variable  terms,  the  jwu-cr  series 

f{z)  =  a„  -f  a^  (z  -  a)  -f  a,^  (z  -  af  +  •  •  •  +  <'n  (-  -  <^")"  +  ■■■       (^) 

is  perhaps  the  most  important.  Here  z,  a,  and  the  coefficients  </,-  may 
be  either  real  or  complex  numbers.  This  series  may  be  written  more 
simply  by  setting  x  =  z  —  a;  then 

fix  +  a)  =  <t>  (x)  =  a^  +  a^x  +  o^:r  +  •  •  •  +  ''„.'•"  +  •  •  •  (<)') 

is  a  series  which  surely  converges  i'oi'  x  =  0.  It  may  or  may  not  con- 
verge for  other  values  of  x,  but  from  Ex.  15  or  19  above  it  is  seen 
that  if  the  series  converges  for  A',  it  converges  absolutely  for  any  x 
of  smaller  absolute  value ;  that  is,  if  a  circle  of  radius  A'  be  drawn 
around  the  origin  in  the  complex  plane  for  x  or  about 
the  point  a  in  the  complex  plane  for  z,  the  series  (9) 
and  (9')  respectively  will  converge  absolutely  for  all 
complex  numbers  which  lie  Avithin  these  circles. 

Three  cases  should  be  distinguished.  First  the 
series  may  converge  for  any  value  x  no  matter  how 
great  its  absolute  value.  Tlie  cii-cle  may  tlien  have 
an  indefinitely  large  radius ;  the  series  converge  for  all  values  of  x  or  ,-;; 
and  the  function  defined  by  them  is  Unite  (whether  real  or  com])lex) 
for  all  values  of  the  argunu'nt.  Such  a  function  is  called  an  integnd 
function  of  the  complex  varial)le  z  or  ,/•.  Secondly,  the  series  may  con- 
verge for  no  other  value  than  x  =  0  or  z  =  (t  and  therefoi'e  cannot  define 
any  function.  Thirdly,  thei'c  may  be  a  definite  largest  value  for  the 
radius,  say  R,  such  that  for  any  ])oint  within  the  respective  circles  of 
radius  R  the  series  converge  and  define  a  function,  whereas  for  any  ])oint 
outside  the  circles  the  series  diverge.  The  circle  of  radius  R  is  calU'd 
the  circle  of  convergence  of  the  series. 

x\s  the  matter  of  the  radius  and  circle  of  convergence  is  important,  it  will  be 
well  to  go  over  the  whole  matter  in  detail.    Consider  the  suite  of  nund)ers 

Irtjl,  v|«o|,  "^''V'sl,  ■••i  ^'V^n\-  •■■■ 

Let  them  be  imagined  to  be  located  as  points  with  coordinates  between  0  and  +  co 
on  a  line.    Three  possibilities  as  to  the  distribution  of  the  poiuls  ai'isi'.    First  they 


4^4  THEORY  OF  FUNCTIONS 

may  be  unlimited  above,  that  is,  it  may  be  possible  to  picl<:  out  from  the  suite  a  set 
of  numbers  which  increase  without  limit.  Secondly,  the  numbers  may  converge  to 
the  limit  0.  Thirdly,  neither  of  these  suppositions  is  true  and  the  mimbers  from  0 
to  +  CO  may  be  divided  into  two  classes  such  that  every  number  in  the  first  class  is 
less  than  an  infinity  of  numbers  of  the  suite,  whereas  any  number  of  the  second 
class  is  surpassed  by  only  a  finite  number  of  the  numbers  in  the  suite.  The  two 
classes  will  then  have  a  frontier  luunber  which  will  be  represented  by  1/R 
(see§§19ff.). 

In  the  first  case  no  matter  what  x  may  be  it  is  possible  to  pick  out  members 
from  the  suite  such  that  the  set  v | aj \ ,  v |a/|,  Vlai-I, •  •  ■ ,  with  i  <j <  k-  •  • ,  increases 
without  limit.  Hence  the  set  v|a,||x|,  V\ aj |  |x|,  •  •  •  will  increase  without  limit ;  the 
terms  OiX^  dj^,-  •  ■  of  the  series  (9')  do  not  approach  0  as  their  limit,  and  the  series 
diverges  for  all  values  of  x  other  than  0.  In  the  second  case  the  series  converges 
for  any  value  of  x.  For  let  e  be  any  number  less  than  l/|x|.  It  is  possible  to  go  so 
far  in  the  suite  that  all  subsequent  numbers  of  it  shall  be  less  than  this  assigned  e. 
Then 

\an+pX"+P\<€''+P\x\"+P     and     e"|x|»+ e"+i|x|«  +  i  +  •  •  ■,  e|x|<l, 

serves  as  a  comparison  series  to  insure  the  absolute  convergence  of  (9').  In  the 
third  case  the  series  converges  for  any  x  such  that|x|<  R  but  diverges  for  any 
X  such  that  |  x |  >  A*.  For  if  |  x  |  <  7?,  take  e  <  /?  —  |  x  |  so  that  |  x  |  <  1!  —  e.  Now  proceed 
in  the  suite  so  far  that  all  the  subsequent  niimbers  shall  be  less  than  l/(/i  —  e), 
which  is  greater  than  \/R.   Then 

|a„  +  „x«+^M  <  ^^ <  1,     and     V  -^-^ 

0  ' 

will  do  as  a  comparison  series.  If  |x|  >  R,  it  is  easy  to  show  the  terms  of  (9')  do  not 
approach  the  limit  0. 

Let  a  circle  of  radius  r  less  than  7?  be  drawn  concentric  with  the 
circle  of  convergence.  Then  vifliin  the  circle  of  radlK.^  r  <  li  the  poire 
KCfics  (0')  conrerr/cs  unifoinnhj  and  defines,  a  continuous  fiincfia)}  :  ilie 
Integral  of  the  function  niaij  he  had  hy  intcfj rating  the  series  term  hij 
term, 

r^  11  1 

cE>  (.t)  =  I     ^  (x)  dx  =  a^x  +  ~  a/-  +  r.  ''./"^  +•■•  +  --  "„_,•''"  +  •  •  • ; 

and  the  series  of  deriratires  coni'erges  iiniformhj  and  rejiresents  th^ 
derivative^  of  the  function, 

<^'(.r)  =  <,^-\-2  a_4r  -f  3  a^x'^  -\ h  7i//,,/'»-i  -\ . 

To  prove  these  theorems  it  is  merely  necessaiy  to  set  up  an  .U-series 
for  the  series  itself  and  for  the  series  of  derivatives.  Let  A'  be  any 
number  between  r  and  11.    Then 

1^1  +  l"i'-^'  +  l"J-^"'  +  •  ■  •  +  !"„|  A'"  +  •  ■•  (10) 


INFINITE  SERIES  435 

converges  because  A'  <  R  ;  and  furthermore  {('nX"]  <  |i'^,j  A'"  holds  for  any 
X  such  that  |  .r  |  <  A',  that  is,  for  all  points  within  and  on  the  circle  of 
radius  r.    Moreover  as  |.r|  <  A, 

\nanX"-^\  =  \a„\^(^y\x"<\a„\X'^ 

holds  for  sufficiently  large  values  of  n  and  for  any  x  such  that  |a?|  =  r. 
Hence  (10)  serves  as  an  ^/-series  for  the  given  series  and  the  series  of 
derivatives  ;  and  the  theorems  are  proved.  It  should  be  noticed  that  it 
is  incorrect  to  say  that  the  convergence  is  uniform  over  the  circle  of 
radius  R,  although  the  statement  is  true  of  any  circle  within  that  circle 
no  matter  how  small  R  —  r.  For  an  apparently  slight  but  none  the 
less  important  extension  to  include,  in  some  cases,  some  points  upon 
the  circle  of  convergence  see  Ex.  5. 

An  immediate  corollary  of  the  above  theorems  is  that  any  power 
sei'les  (9)  in  the  complex  variable  which  converges  for  oilier  values  than 
z  =  a,  and  hence  has  a  finite  circle  of  convergence  or  conrerges  all  over 
the  cornpjlex  pjlane,  defines  an  anal ijtic  function  fiz)  of  z  in  the  sense  of 
§§  73,  126;  for  the  series  is  differentiable  within  any  circle  within  the 
circle  of  convergence  and  thus  the  function  has  a  definite  finite  and 
continuous  derivative. 

167.  It  is  now  possible  to  extend  Taylor's  and  ]\raclaurin's  Formulas, 
which  developed  a  function  of  a  real  variable  x  into  a  polynomial  plus 
a  remainder,  to  infinite  series  known  as  Taylor's  and  Maclaurin's  Series, 
which  express  the  function  as  a  power  series,  provided  the  remainder 
after  n  terms  converges  uniformly  toward  0  as  n  becomes  infinite.  It 
Avill  be  sufficient  to  treat  one  case.    Let 

/■(■■■)  =/(0)  +/'lO),.-  +  j^./'"(0)..-=  + . . .  +  ^.^^J^^^,/'— '(O).---  +  /.'„, 
lim  R„{x)  =  0  uniformly  in  some  interval  —  h  ^  x  ^  h, 

n  =  » 

where  the  first  line  is  ^laclaurin's  Formula,  the  second  gives  differnet 
forms  of  the  remainder,  and  the  third  expresses  the  condition  that  the 
remainder  converges  to  0.    Then  the  series 

/(0)+/'(0).T  +  |,/"(0)cc-^ 

+  ■  •  •  +  (,rhy^/''-'^(:o)-^-^  +  ;^/^'"(0)^"  +  •  •  •     (11) 


486  THEORY  OF  FU^X'TIOXS 

converges  to  tlie  value /(./■)  for  any  x  in  the  intervnl.  The  proof  con- 
sists merely  in  noting  that _/'(./■)  —  Il„(x)  =  ^'„(^')  is  the  sum  of  the  first 
n  terms  of  the  series  and  that  |yi„(x)|  <  e. 

In  the  case  of  the  exponential  function  e^  the  ?ith  derivative  is  e^,  and  the  re- 
mainder, taken  in  the  lir.st  form,  becomes 

2?„  (x)  =  —  eS-'x",        I  Ru  (^)  I  <  —  fe''/i".         I J I  ^  /i. 
n\  ?(  ! 

As  n  becomes  inlinite,  /.'„  clearly  approaches  zero  no  matter  what  the  value  of  h  ; 

^^^  a-2        c3  ,„ 

e'-  =  1  +  X  +  —  +  —  4-  •  •  •  +  —  +  •  •  • 
2  13!  n I 

is  the  infinite  series  for  the  exponential  function.    The  series  converges  for  all 

values  of  x  real  or  complex  and  may  be  taken  as  the  definition  of  e^  for  complex 

values.  This  definition  may  lie  shown  to  coincide  with  that  obtained  otherwise  (§  74). 

For  the  expan.sion  of  (1  -i-  x)'"  the  remainder  may  be  taken  in  the  second  form. 

i?„(x)  =  ^»('H-i)---(»^-n  +  i)^„  .j_-^y-i  _ 

, -n   ,  >,      \m(m  —  1)  ■  ■  ■  (m  —  n  +  l)\ -,    ,,       ,,       ,  ,       , 

j  1.2---(H-1)  I 

Hence  when  h  <1  the  limit  of  /i'„  (x)  is  zero  and  the  infinite  expansion 

niim  —  l)    .,      m(m  — l)(m  — 2)    „ 

(1  +  x)'"  =  1  +  mx  +  — '  x'-i  +  — ^ '-^ '-  x3  +  .  . . 

2  :  3  ! 

is  valid  for  (1  +  x)">  for  all  valut's  uf  x  imnu'rically  less  than  tuiity. 
If  in  the  binomial  expansion  ,c  be  replaced  by  —  x'^  and  7u  bj^  —  .', , 

1  ,       1    ,      1  •  3    ,       1  •  3  •  5   ,       1  •  3  •  5  •  7   , 

1  +       X-  H X-*  + X6  +  -^ X8  -I-   •  .  .  . 


Vl  _  x2  -  ^  ■  -1  2  ■  4  .  0  2  •  4  •  0  •  8 

Tills  series  converges  for  all  values  (jf  x  mimerically  less  than  1.  and  hence  con- 
verges uniforndy  whenever  |X|^  h  <  1.    It  may  therefore  be  integrated  term  by 

^^'™'-  .      ,  1  x3      1  .  3  x'^       1  •  3  •  5  X"       1  •  3  •  5  •  7  x9 

23       2-45       2.4.67       2.4.U.8  9 

This  series  is  valid  for  all  values  of  x  ntunerically  less  than  unity.    The  series  also 
converges  for  x  —  ±  1 ,  and  hence  by  Ex.  5  is  uniformly  convergent  when  —  1  =  x  s  i . 

I)Ut  Taylor's  and  ]\raclaurin's  series  may  also  l)e  extended  directly  to 
fun(;tions /('-.■)  of  a  com])lex  variahle.  If  _/'(-')  is  single  valued  tuid  has 
a  definite  continuous  derivative /''(,^)  at  every  point  of  a  region  and  on 
the  boundary,  the  expansion 

f(z)  =  f(a)  +  f'(a)  (z  -  a)  +  ...+/<«-  ^ )  (a)  -J-~^]^y     +  n^ 
has  Ix'cn  established  (§  ]2())  with  tlu-  rcnuiindcr  in  tlic  form 

2  TT       J,,  (f  —  (ly  (t  —  Z)  ^"^   2lT  p"  p  —  /• 


l^^»(-)l 


INFINITE  SERIES  487 

for  all  points  z  within  the  circle  of  radius  r  (Ex.  7,  p.  306).  As  n  becomes 
infinite,  7i„  approaches  zero  uniformly,  and  hence  the  infinite  series 

m  =  /(«)  +  /'(«)  (^  -  a)  +  . . .  +  /<"^  (a)  ^^^~  +  •  •  •        (12) 

is  valid  at  all  points  within  the  circle  of  radius  r  and  upon  its  circum- 
ference. The  expansion  is  therefore  convergent  and  valid  for  any  z 
actually  within  the  circle  of  radius  p. 

Even  for  real  expansions  (11)  the  significance  of  this  result  is  great 
because,  except  in  the  simplest  cases,  it  is  impossible  to  compute  /'"^  (,r ) 
and  establish  the  convergence  of  Taylor's  series  for  real  variables.  The 
result  just  found  shows  that  if  the  values  of  the  function  be  considert'd 
for  complex  values  z  in  addition  to  real  values  a-,  the  circle  of  ctonvei- 
gence  will  extend  out  to  the  nearest  point  where  the  conditions  imposed 
on  f{z)  break  down,  that  is,  to  the  nearest  point  at  which  fiz)  becomes 
infinite  or  otherwise  ceases  to  have  a  definite  continuous  derivative/' (.t). 
For  example,  there  is  nothing  in  the  behavior  of  the  function 

(1  +  a:-)- 1  =  1  —  X-  +  x^  —  j^  -\-:)^ , 

as  far  as  real  values  are  concerned,  whidi  should  indicate  why  the  expan- 
sion holds  only  when  \x\  <  1 ;  but  in  the  complex  domain  the  function 
(1 +  '-')"  ^  becomes  infinite  at  z=  ±  /,  and  hence  the  greatest  circle 
about  ;s;  =  0  in  which  the  series  could  be  expected  to  converge  has  a  unit 
radius.  Hence  by  considering  (1  -f-  ,t-)^^  for  complex  values,  it  can  be 
predicted  without  the  examination  of  the  n\X\  derivative  that  the  Mac;- 
laurin  development  of  (1  +  .r-)-^  will  converge  when  and  only  Avhen  x. 
is  a  proper  fraction. 

EXERCISES 

1.  (a)  Does  x  +  a;  (1  —  a;)  +  x  (1  —  x)'^  +  •  •  •  converge  uniformly  when  0  ^  x  s  i  ? 

1  2 ^      CI  _  M  n 2  lis 

(/3)  Does  the  series  (1  +  A:)i-  =  1  +  1  + f-  ^^ '-^ -'  +  . . .  converge  uni- 

fcn-mly  for  small  values  of  k  ?  Can  the  derivation  of  the  limit  e  of  §  4  thus  be  made 
rigorous  and  the  value  be  found  by  setting  ):  =  0  in  the  series '? 

2.  Test  these  series  for  uniform  convergence  ;  also  the  series  of  derivatives-. 

{a)  1  +  X  sin  0  ^  x-  sin  2  (9  +  x^  sin  3  6*  +  •  •  • ,  |x  |  ^  X  <  ], 

,^^^   _    ,   sinx      sin^x      sin^x       sin''x  ,    , 

\^1  ^■^i  .3-2  4-5 

^     ,     X  -  1  1   /X  -  1\2  1  /x  -  1\3  1  _        _    -,, 

X  2\x/        3\x/  2 

,^,    X-1  1    /X-l\3         1   /x-]\" 

(5) h  -    + -H  •  •  • .  0  <  7  =  X  ==  X  <  CO  . 

^       X  +  1      3  \x  +  1/        5  \x  +  1/  /  -      - 

(e)  Consider  complex  as  well  as  real  values  of  the  variable. 


438  THEORY   OF  FUNCTIONS 

3.  Determine  the  radius  of  convergence  and  draw  the  circle.   Note  that  in  prac- 
tice the  test  ratio  is  more  convenient  than  tlie  tlieoretical  method  of  tlie  text: 

(a)  X  -  l  X-  +  i  j-3  -  i  j-<  +  •  •  • ,  03)  X-  Ix^  +  ix''-  Ix'  +  ■■■, 

a\_         a        a-         a^  J  2  13  14! 

( e  )   1  X  -  (^  +  I) X-  +  (1  +  ^  +  ^) ^-3  -  (i  +  I  +  i  +  i)x*  +  .  .  . , 

,    ,    ,       32  +  3    ,       3*  +  3    ,       3"  +  3    „ 

^^'  4.21  4.4!  4.G1  ' 

{■,))   1  -  X  +  X*  -  X-'  +  ./•?  -  ,/^  +  x^-  -  x^^  +  •  ■  • , 

(^)     (X-l)l-    \(x-\f-\-    >(,._  1)3  _!(,._  1)4  +  ...^ 

_  (»i-l)(m  +  -2)  _^,.  _^  (m-1)(m-3)(m  +  2)(»i  +  4)^, ^ 

3  1  5  ! 

(k)    1- 


22(7M  +1)       2*  •  2  !  (til  +  1)  (»i,  +  2)       2»  •  3  1  (mi  +  1)  (m  +  2)  (»t  +  3) 

X-  x*      /I       1\  x6      /I       1       1\  x8      /I       1       1       1\ 

<^>2^-2M^(t  +  2)  +  2;H3^  (1  +  2 +  3)-2^(4Tj^  (1+2  +  3  +  4)  +••■• 

,    ,   ,  ^   a^  ,,   a(a +  l)^(/3  +  l)  fl:(a  +  1)  (a  +  2)^(/3 +  1)  (/3  +  2) 

1  u )    I  A X  H X-  +   X*^  +  •  •  •  . 

1-7  1-2. 7(7  +  1)  1.2.3.7(7  +  1)(7  +  -^) 

4.  Establish  the  Maclaurin  expansions  for  the  elementary  functions: 

(a)  log  (1  —  x),         (/3)  sinx,  (7)  cosx,  (5)  cosh  x, 

(e)  a^,  (f)  tan-'ix,         (17)  sinh-ix,         {6)  tanh-ix. 

5.  AbeVs  Theorem.  If  the  infinite  series  a^  +  a^x  +  rtoX-  +  u..x^  +  •  •  ■  converges 
for  the  value  X,  it  converges  uniformly  in  the  interval  0  ^  x  ^  A'.  Fnive  this  b}^ 
showing  that  (see  Exs.  17-19,  p.  428) 

I  l',M)  I  =  1  anX'^  +  cin  +ix»  +1  +  .  .  .  I  <  /^Vl  rt„A'''  +  •  ■  •  +  a„  ^  „ A-«  +v'  I. 

when  J)  is  rightly  chosen.  Apply  this  to  extending  the  interval  over  which  the 
series  is  uniformly  convergent  to  extreme  values  of  the  interval  of  convergence 
wherever  possible  in  Exs.  4  (cr),  (f),  {6). 

6.  I'^xamine  sundry  of  the  series  of  Ex.3  in  regard  to  their  convergence  at  ex- 
treme points  of  the  interval  of  convergence  or  at  various  other  points  of  the  circum- 
ference of  their  circle  of  convergence.    Note  the  significance  in  view  of  Ex.  .j. 

_  1 

7.  Show  that /(x)  =  e   2--./(0)  =  0.  cannot  be  expanded  into  an  inliniti'  Mac- 

_2_ 
laurin  series  by  showing  that  A',,  =  e    ^-,  and  hence   that   /i'„   docs  nut    cnnvcr-c 
uniformly  toward  0  (see  Ex.9,  p.  (30).    Show  this  also  from  tlie  coiisiilcratinn  (,f 
complex  values  of  x. 

8.  From  the  consideration  of  complex  values  determine  the  interval  of  con- 
vergence of  the  Maclaurin  series  for 

(a)  tanx  =  ^^^,         (/3)  ^— ,         (7)  tanh  x.         (5)  lo-(l  +  c-'). 
cos  X  e^  —  1 


r<l 


INFINITE   SERIES  439 

9.  Show  that  if  two  .siinihir  infinite  power  series  represent  the  same  function 
in  any  interval  the  coefficients  in  the  series  must  be  equal  (cf.  §  32). 

10.  From  1  +  2  r  cos x  +  r-  =  (1  +  r&^)  (1  +  re-  ")  =  r^  M  +  —  j  /i  +  ^^ 

/  r-  r^  \ 

Ijrove         log  (1  +  2  r  cos  x  +  }-'^)  =  2  /  r  cos  x—  —  cos  2  j:  +  —  cos  3  x  —  •  •  • )  , 

XX  /  J.2  ,.3  \ 

log  (1  +  2  /•  cos  J"  +  7-"-^)  t/x  =  2  I  /•  sin  x  —  —  sin  2  x  +  ~  sin  3  x  —  •  •  •  1 

11/1,,  ,     ox       ,1  ,  nA'osx      cos2x      cos3x  \ 

anil      log  (1  +  2  r  cos  x  +  r-)  =  2  log  r  +  2 i .  ■  . ) 

\    r  2  r-  3  r-  / 

)■  >  1 
r'l      /I    ,    -1  ,•-.,,         n     ,  ,   „ /sin  X       sni2x      sin3x  \ 

hi;,^  (1  +  2  r  cos  x  +  r-)  tZx  =  2  x  log  r  +  2  ( 1 •  •  •  I  • 

J,j       ■   '  \    y  2-  /•-  3-  r^  / 

I      loii  (1  +  sni  tr  cos  x)  ax  =  2  x  log  cos  — h  2  |  tan  —  sm  x  —  tan-  —  ■ 1- 

Jf>  '^  2  \         2  2       2--2 

1 1     „  r^      dx  ^         1,1.3  ]  ■  3 .5  /> '      ^Zx 

11.  Prove   /     —  =1 1 v.  .  .  .  =  \     

-'u    ^    1  +  ^4  2-5       2  ■  4  .  !)       2  .  4  .  «i .  13  -'i     a  1  +  j^ 

12.  Evaluate  these  integrals  by  expansion  into  sci'ies  (see  Ex.  23.  p.  4.")2) 

((r)     /       dx  =  -~-(-]  +^J-] =  tan-i-, 

J{)  x 


(/3)  ^(Zx  =  7rsui-iA:,  (7) —  cZx  =  —  , 

•^1)  cosx  v'li    1  +  cos^x  4 

J^  °°  Vtt   -  ( -  ^'  r  '^ 

e-  '^■^"  cos  2  /3xtZx  = e    V^y  ^  (e)     |     ],jo-  (i  +  2  r  cos  x  +  r") 
u                                        2  a  Jo 


13.  By  formal  nuiltiplication  (^  168)  show  that 
1-  a- 


1  —  2  a  cos  X  +  a^ 
a  sin  X 


-  =  1  +  2  or  cos  X  +  2  a-  cos  2  x  +  •  •  • , 
a  sin  X  +  a-  sin  2x  +  •  ■  •  . 


1  —  2  a:  cos  X  +  cr- 

14.  Evaluate,  by  use  of  Ex.  13.  these  detinite  integrals,  m  an  integer: 

/"^         cos?nxr7x  tto""  ,  ,     r'^         x  sin  xdx  tt, 

{a) = ,  (/3) =  -  log(l  +  a). 

Jo    1  — 2  or  cosx  +  a-       1  —  a-  Ju    1  — 2acosx  +  a^       a 


C'"      sm 

Jo     1  —  2 

1^0  (1 — i 


a  cos  X  +  a-       2 
sin^xrix 


(1  —  2a  COSX+  a-)(l-— 2/3 cosx  +  /S^) 

15.  In  Ex.  14  (7)  let  a  =  1  —  h/m  and  x  =  ^/??i.    Obtain  by  a  limiting  process, 
and  by  a  similar  method  exercised  upon  Ex.  14  (a) : 

z' -^  z  sin  zfZz      TT      ,  c^  <io%z(lz      tt 

Jo         /(2  +  22  2  Jo        k-  +  2^  2 

Can  the  u.se  of  these  limiting  processes  be  readily  justified  ? 


440  THEORY  OF  FUNCTIONS 

16.  Let  h  and  x  be  less  than  1.    Assume  the  expansion 

/(x,  h)  =  ^  =  1  +  /iPi(x)  +  h^P^x)  +  ■■■  +  /i«P„(x)  +  . . . « 

Vl  -  2  xh  +  /i^ 

Obtain  therefrom  the  following  expansions  by  differentiation  : 

-X  = ^- z=P'i  +  '^^2  +  f'^'P's  +  ■■■  +  f^"-'K  +  •■■, 

"-  (l-2x/i  + /i2)2 

/,:  = ^— '^ -  Pi  +  2  hP„  +  3  IfiP^  +  •  •  •  +  n/i»-iP„  +  .  . .  . 

(l-2x/i  +  /i^)'- 
Hence  establish  the  given  identities  and  consequent  relations: 

?J^V;  =       xp;  +  /I  (xp;  _  p;)  + . . .  +  /,« ~i(xp;  -  p;,  _,)      + . . .  = 

^'-^x-/^  - 1  +  p;  +  /K^;  -  -Pi)  +  •  •  •  +  /^"(p:.+i  +  p;-i  -  Pn)  +  •  •  •  = 

2  X///  =  /t  (2  X)  +  •  ■  •  +  /*«(2  -^Pn  -i)  • 

Or         nPn  =  xp;  -  p;  _i  and  7^;  ^^  +  p;  _i  -  P„  =  2  xp; . 

Hence        xP^  =  P;  +i  -  (7i  +  1)  P„     and     (x^  -  1)  P;  =  ji  (xP„  -  P„  _i) . 

Compare  the  results  with  Exs.  13  and  17,  p.  252,  to  identify  the  functions  with  the 
Legendre  polynomials.    Write 

1 1 1 

(1-2  xh  +  ffi)i      (1  -  2  /i  cos  0  +  Ifi)^      (1  -  he'^y^  (1  -  he-  ''«)^ 

=  f  1  +  1  heie  +  l^  Jfie^  !8  +  . .  .\  /i  +  ^  he-  ie  +  111  h"~e-'^'e  +■■■], 
\        2  2-4  A        2  2-4  / 

and  show  P„(cos  6)  =  2  Ll^;  "  '  (^ '^  -  ^)  |  ^^^  ,,^  ^ Ll"^ cos (,i  _  2)  ^  +  .  .  .  | . 

2  •  4  •  •  •  2  ?i        (^  1  •  (2n  —  1)  J 

168.  Manipulation  of  series.   If  an  Infinite  series 

S  =  >f^  +  »i  +  v^,  +  •  ■  •  +  "„_i  +  7/„  4-  •  •  •  (13) 

converges,  the  serifs  obtained  bi/  (/roKpiruj  tlie  terms  in  jxirentlieses  icitli- 
out  altering  tJicir  order  if  ill  also  converge.    Let 

,S'  =  ^/^  +  r/,  +  . . .  +  U„_,  +  f/,.  +  ■  ■ .  (13') 

and  .s'l,  S'j,,  •  ■  ■ ,  S',^.,  ■  ■  ■ 

be  the  neAv  series  and  the  sums  of  its  hrst  71'  terms.  These  sums  are 
mert'ly  particuhir  ones  of  the  set  S^,  S,„  ■■■,  S,^,---,  and  as  71'  <  71  it 
follows  that  71  becomes  infinite  when  ?i'  does  if  71  be  so  chosen  that 
.S'„  =  S'^,.  As  S^  ap])roaches  a  limit,  S'„,  must  a])pi'oach  the  same  limit. 
As  a  corollary  it  a^jpears  that  if  the  series  obtained  by  removing-  ])aren- 
theses  in  a  given  series  converges,  the  value  of  the  series  is  not  affected 
by  removing  the  parentheses. 


INFINITE  SERIES  441 

If  two  convergent  infinite  series  he  glcen  as 

S  =  «o  +  ?/j  +  •  ■  • ,     (md      T  =  v^  +  I'l  H , 

then  (Xu^  +  fxv^)  +  (A»^  +  fJ.>\)  -\ 

trill  ronrerge  to  tlie  limit  XS  +  /^'I,  (^nd  ic ill  converge  cihsolatel tj provided 
hotli  the  given  series  converge  ahsolutel ij .  The  proof  is  left  to  the  reader. 
If  (I  given  series  converges  ahsolutel  ij,  the  series  formed  hij  rearranging 
the  terms  in  anij  order  vithovt  omitting  any  terms  icill  converge  to  the 
same  value.    Let  the  two  arrangements  be 

^  =   ''o  +   "l  +   "j  +  •  •  •  +   ";/  -  1  +   "«  +  •  •  ■ 

and  S  =  u^^,  +  u^,  +  ;/._,,  +  •  •  •  +  ",,'-1  +  ''„-  +  •  •  •  • 

As  .S'  converges  ahsolutely,  ?i  may  be  taken  so  large  that 

and  as  the  terms  in  S'  are  identical  with  those  in  S  except  for  their 
order,  n'  may  be  taken  so  large  that  S'„,  shall  contain  all  the  terms  in 
.S'„.  The  other  terms  in  S',^,  will  be  found  among  the  terms  ?/„,  ?/„^.i,  •  •  ■• 
Hence  , , ,.         r.  1  ^  1      1,1  1   , 

As  |.S'  —  .S',,]  <  e,  it  follows  thatj.V  —  .S',',,j  <  2  e.  Henct^  ,S^,  approaches  .S' 
as  a  limit  when  71'  becomes  infinite.  It  may  easily  be  shown  that  S'  also 
converges  absolutely. 

77ie  tlieorem  is  still  triw  if  the  rcarrangfinent  ofS  is  into  a  series  some 
of  u-hose  terms  are  thenisclvcs  infinitt'  srrirs  of  terms  selected  from   S. 

where  f/,-  may  be  any  aggregate  of  terms  selected  from  S.  If  U^  be  an 
infinite  series  of  terms  selected  from  S,  as 

f"-  =  ''m  +  '',-1  4-  i'ii  H h  ",■„  H , 

the  absolute  convergence  of  L'-  follows  from  that  of  S  ('cf.  Ex.  22  below). 
It  is  possible  to  take  n'  so  large  tliat  ewvy  term  in  .s'„  shall  occur  in  one 
of  the  terms  U^,  l\.  ■  ■  ■,  U,^,_-^.    Then  if  from 

s-  r^-  i\ u„_,  (14) 

there  be  canceled  all  the  terms  of  .S„,  the  terms  which  remain  will  be 
found  among  u,^,  ?^, +  1,  •••,  and  (14)  will  be  less  than  e.  Hence  as  n' 
becomes  infinite,  the  difference  (14)  approaches  zero  as  a  limit  and  tlie 
theorem  is  proved  that 

S=L\+  L\-\-...+  Lv-i  +  U,,  +  ...  =  S'. 


442  THEOEY  OF  FUNCTIONS 

If  a  series  of  real  terms  is  convergent,  but  not  absolutely,  the  number  of  posi- 
tive and  the  number  of  negative  terms  is  infinite,  the  series  of  positive  terms  and 
the  series  of  negative  terms  diverge,  and  the  given  series  may  be  so  rearranged  as 
to  comport  itself  in  any  desired  manner.  That  the  number  of  terms  of  each  sign 
cannot  be  finite  follows  from  the  fact  that  if  it  were,  it  would  be  possible  to  go  so 
far  in  the  series  that  all  subsequent  terms  would  have  the  same  sign  and  the  series 
would  therefore  converge  absolutely  if  at  all.  Consider  next  the  sum  S„  =  Pj—  X,„, 
/  4-  ??i  =  n,  of  n  terms  of  the  series,  where  Pi  is  the  sum  of  the  positive  terms  and 
Nm  that  of  the  negative  terms.  If  both  Pi  and  y„,  converged,  then  Pi  +  X,„  would 
also  converge  and  the  series  would  converge  absolutely ;  if  only  oue  of  the  sums 
Pi  or  Nm  diverged,  then  S  would  diverge.  Hence  both  sums  must  diverge.  The 
series  may  now  be  rearranged  to  approach  any  desired  limit,  to  become  positivelj' 
or  negatively  infinite,  or  to  oscillate  as  desired.  For  suppose  an  arrangement  to 
approach  I,  as  a  limit  were  desired.  Fir.st  take  enough  positive  terms  to  make  the 
sum  exceed  i,  then  enough  negative  terms  to  make  it  less  than  i,  then  enough 
positive  terms  to  bring  it  again  in  excess  of  L.  and  so  on.  But  as  the  given  series 
converges,  its  terms  approach  0  as  a  limit ;  and  as  the  new  arrangement  gives  a 
sum  which  never  differs  from  L  by  more  than  the  last  term  in  it.  the  difference 
between  the  sum  and  L  is  approaching  0  and  L  is  the  limit  of  the  sum.  In  a  sinnlar 
way  it  could  be  shown  that  an  arrangement  which  would  comport  itself  in  any  of 
the  other  ways  mentioned  would  be  possible. 

If  two  absolufeJ//  ronrerr/ent  series  he  viultq/I'ied,  as 

^  =    "o  +    >'l   +    "-2^ +    "„   H , 

T=  r, +  /•!  +  r, +  --.  +  r„  +  ..., 

and  W  =  ii„i\,  +  i/^r^,  +  I/.,!-,,  +  •  •  •  +  i'-„'\,  -i 

+  ''o''i  +  "i'\  +  if-2'\  +  •  •  •  +  <'„>\  +  •■■ 
+ 

+    "o''«  +    "l'-„  +    t'-2'-n  H +    "„'■„  H 

+ 

and  Iff/te  terms  In  W  hi'  arrawjed  In  a  slmph-  series  as 

"o'"o  +  ("i''o  +  ''i''i  +  "o'"i'>  +  (>'.''o  +  "■>J\  +  ".''i  +  "i''^  +  "o''.>  +  ■  •  • 
or   in  (inij  nflirr  rn<inncr  (rlidtsoi'ri'r.  the  serirs   is  ifhsoh/fr/i/  i-irari'i'fjent 
and  eonrerges  to  tlie  rohic  of  tlie  product  ST. 

In  the  particular  arrangement  above.  S^T^,  <s'.,7'.,-  -S, '/'„  i'^  the  sum  of 
the  tirst,  the  tirst  two,  the  iirst  n  terms  of  the  series  of  iiarcntlifscs.  As 
lim  .S'„7'„  =  ST,  the  series  of  parentheses  converges  to  S'J'.  As  .S  and  T 
are  absolutely  convergent  the  same  reasoning  could  be  ap})lied  to  the 
series  of  absolute  values  and 

would  be  seen  to  converge.    Hence  the  convergence  of  the  series 


IXFIXITE  SERIES  443 

is  aljsolute  and  to  the  value  .S'7'  when  the  parentlieses  are  omitted. 
Moreover,  any  other  arrangement,  such  in  particular  as 

would  give  a  series  converging  absolutely  to  ,s'7'. 

The  equivalence  of  a  function  and  its  Taylor  or  ]\[aclaurin  infinite 
series  (wherever  the  series  converges)  lends  importance  to  the  operations 
of  multiplication,  division,  and  so  on,  which  may  be  performed  on  the 
series.    Thus  if 

/(,.'•;  =  ''o  +  ('^:r  +  a^-  +  a^j'"^  H ,  \j'\<  I!^, 

0 (:'■)  =  '',  +  ^\-'-  +  ^o^-'  +  ^i'-'  +  ■■■,  I'^' i  <  ^o, 

the  multiplication  may  be  ])erformed  and  the  series  arranged  as 

fC:r)ff(x)  =  aj>^  +  (aj>^  +  nh^)x  +  (aj>,^  +  .//.^  +  aj>^)x'  +  •  •  • 

according  to  ascending  powers  of./-  whenever,/-  is  numerically  less  than 
the  smaller  of  the  two  radii  of  convergence  J!^.  7.'.,,  because  V>oth  series 
Avill  then  converge  absolutely.  ^Moreover,  Ex.  ~>  above  shows  that  this 
form  of  the  product  may  still  be  applied  at  the  extremities  of  its  inter- 
val of  convergence  for  real  values  of  x  provided  the  series  converges 
for  those  values. 

As  an  example  in  tlie  Jiuiltiplication  of  .scries  let  tlie  product  sin  j  cos  jr  lie  found. 

1,1.  ,        1     ,        1     ^        1     „ 

sm  X  =  X x^  -1 X'  —  •  ■  • .         cos X  =  1  —       x'-2  H x'* x^  +  •  ■  ■ . 

3  1  5  1  '  2  1  4  1  6  1 

The  product  will  contain  only  odd  powers  of  x.    The  first  few  terms  are 

/I         1  \    ,       / 1  1  1  \    .       /  1  1  1  1 

1  X- h  ^  l-f^  +    —  + h  ^   J-'  -    —  H h h  — 

\.3I      2  1/  V3  1      3  12  1      4  1/  \7  1       .5 1 2  1      3  1 4  1      6  1 

The  law  of  formation  of  the  coefficients  liives  as  the  cuetficient  of  x-^"  +  i 

,   ,,r     1              1               1           ■         1  1    1 

(- 1)^  + + +  •  ■  •  + +  —  ■■ 

^        'lv^k+})\       (2A--1)121       (2A--3)141  3  1  (2 1  -  2)  1       (2^)lJ 

(-IV-     Pi    ,    (2/.- 4-1)2/.-   ,    (2t+l)(2/.-)(2t-l)(2/,,--2)    ,           ,    (2 A- 4-1) 
_[  H -_-- 1 — h  -  •  -  -1 --. 


(2A--fl)lL  2  1  4  1 

But      2-^'---i  =  (1  +  l)-^-  +  i  =  1  +  (2A-  -fl)  -f  ^-^'-'+'^y-^^  +  ■■■  +  (2k  +  l)  +  1. 

Hence  it  is  seen  that  the  coefficient  of  x-^-i  takes  every  other  term  in  this  synnnet- 
rical  sum  of  an  even  number  of  terms  and  must  therefore  be  eijual  to  half  the  sum. 
The  product  maj'  then  be  written  as  the  series 


sm  X  cosx 


ir  (2.r)3       (2x)5  11..-, 

-    2  X + ...=:-  sm  2  X. 

2L  31     ^     51  J       2 


444  THEORY  OF  FUNCTIONS 

169.  If  a  function /(./•)  be  expanded  into  a  power  series 

f{^-)  =  %  +  (\x  +  ^/,.t'  +  c^''  +  •  •  • ,  \x\<  R,  (15) 

and  if  x  =  a  is  any  point  witliin  the  oii'cle  of  convergence,  it  nxn/  he 
dcslved  to  ti-ansfonii  tlie  t^crics  into  one  v-lilch  proceeds  aeeordlng  to  j/oicci's 
of  (x  —  a)  and  converges  in  a  circle  ahovt  the  point  x  =  a.  Let  t  =  x  —  o:. 
Tlien  x  =  a  -\-  t  and  hence 

x'^  =  a-  +  2  at  +  /-,  ;r^=  a''  +  3  crt  +  3  at:-  +  f,  ■■■, 

/(;r)  =  o^  +  o^  (a  +  0  +  'r  (rr  +  2  at  +  t')  +  ■  ■  ■ .  (If,') 

Since  \a\  <  R,  tlie  relation  |^|  + 1''|  <  R  will  hold  for  small  values  of  t, 
and  the  series  (lo')  will  converge  for  x  ^=\a\-\-\t\.    Since 

is  absolutely  convergent  for  small  values  of  t,  the  parentheses  in  (15') 
may  be  removed  and  the  terms  collected  as 

/(»■)  =  ^  (/)  =  {((^  +  a^a  +  a./r  +  a.^a^  -\ )  +  {o^-\-2  iiji  +  3  n ,(i-  H )t 

+  («.2  +  '"^  ''Z*^  +  •  •  •)  ^'  +  (".  +  •  •  •)  ^'  +  •  •  •; 
or         /(,>•)  =  c^(,>'  -  a)  =  A^  +  A^{x  -  a)  +  .l._^(,.  -  af 

+  .l3(,r -<  +  ...,  (16) 

where  A^,  A^,  A_,,  ■  ■  ■  are  infinite  series  ;  in  fact 

The  series  (16)  in  cc  —  a  will  surely  converge  within  a  circle  of  radius 
72  —  |a|  about  x  —  a;  but  it  may  converge  in  a  larger  circde.  As  a  matter 
of  fact  it  will  (!onvei'ge  within  the  largest  circle  whose  center  is  at  a  and 
within  which  the  function  has  a  definite  continuous  derivative.  Thus 
Maclaurin's  ex})ansion  for  (1  +  .t^)~^  has  a  unit  radius  of  convergence; 
but  the  expansion  about  x  =  l  into  powers  of  x  —  I-  will  have  a  radius 
of  convergence  equal  to  I  V 5,  Avhich  is  the  distance  from  x  =  ^  to  either 
of  the  points  x  =  ±  i.  If  the  function  had  originally  been  defined  by 
its  development  about  x  =  0,  the  definition  would  have  been  valid  only 
over  the  unit  circle.  The  new  development  about  x  —  ^  will  therefore 
extend  the  definition  to  a  considerable  region  outside  the  oi-iginal 
domain,  and  by  re])eating  the  pro(!ess  the  rf>gion  of  definition  may  be 
extended  further.  As  the  function  is  at  each  step  defined  by  a  ])0wer 
series,  it  remains  analytic.  This  process  of  extending  the  definition  of 
a  function  is  called  anulijtic  continuation. 


INFINITE  SEEIES  445 

Consider  the  expansion  of  a  function  of  a  function.    Let 
f{x)  =  Cq  +  (i^x  +  a^  +  a^x^  H [.^'l  <  7.'^, 

x  =  <i>(y)  =  \  +  h^j  +  ^>.y  +  hy  +  ■■■,       \y\  <  n,, 

and  let  \h^\  <  R^  so  that,  for  sufficiently  small  values  of  y,  the  point  x 
will  still  lie  within  the  circle  B^.  By  the  theorem  on  multiplication,  the 
series  for  x  may  be  squared,  cubed,  •  •  • ,  and  the  series  for  x'^,  o-^,  ■  ■  ■  may 
be  arranged  according  to  powers  of  ?/.  These  results  may  then  be  sub- 
stituted in  the  series  for/(,r)  and  the  result  may  be  ordered  according 
to  powers  of  ij.  Hence  the  expansion  for /[</>(?/)]  is  obtained.  That 
the  expansion  is  valid  at  least  for  small  values  of  1/  may  be  seen  by 
considering 

i  =  K^ol  +  IMI//I  +  Mu?  +  ■■■,         \u\  small, 

which  are  series  of  positive  terms.  The  radius  of  convergence  of  the 
series  for /[</>(,/)]  may  be  found  by  discussing  that  function. 

For  example  consider  the  problem  of  expandijig  e=°«-^  to  five  temis. 
eV  =  1  +  7/  +  ^  ?/2  +  1  y3  +  ^^yi  ^  .  .  .^  y^  cosx  =  1  _  J  J-2  4.  ^i^a;4  ^.  . .  .^ 

?/2  =  i_a;2  4.  1^4 ^      2/^:=l_  1^2+  ■  a-t ,      ?/*  =  1- 2a-2  +  l|-x* , 

ey  =  l  +  {l-lx"-  +  r^^z^ )  +  ^ (1  _  x2  +  ix^ )  +  HI  -  I -i'^  +  1  ic^ ) 

+  ,i^(l-2a-2  +  lfx^ )  +  ■•• 

=  (1  +  1  +  1  +  4  +  .V  +  .  •  •)  -  (i  +  A  +  i  +  A  +  •  •  Oa-' 

+  (i  +  J  +  A  +  i  + •■•)•»*  +  ••-, 

(.y  —  geoex  —  21 1  _  1^  x^  +   |  =  X'*  —  •  •  •  . 

It  .should  be  noted  that  the  coefficients  in  this  series  for  e'^"*-^  are  really  infinite 
series  and  the  final  values  here  given  are  only  the  approximate  values  found  by 
taking  the  first  few  terms  of  each  series.  This  will  always  be  the  case  when 
y  z=  b^  +  h^z  +  •  •  •  begins  with  &q  ^i  0  ;  it  is  also  true  in  the  expansion  about  a  new 
origin,  as  in  a  previous  paragraph.  In  the  latter  case  the  difficulty  cannot  be 
avoided,  but  in  the  case  of  the  expansion  of  a  function  of  a  function  it  is  some- 
times possible  to  make  a  preliminary  change  which  materially  simplifies  the  final 
result  in  that  the  coefficients  become  finite  series.    Thus  here 

gcosx  =  ei  +  2  i=ee-,         z  =  cosx  —  1  =—  ^  a-2  ^  ^1^3.4  _  ,i-xs  +  •  •  •, 

e*  =  1  +  (_  1  x2  +  ,i^x*  -  ,1^x6  +  •••)  +  !  (K  _  ^1^x6  +•••)  +  H-  i-^'  +  •••)  +  ••• , 
gcosx  =;  egj  -  e (1  _  I  x2  4-  ^  X*  —  /^V  ^^  -^ )• 

The  coefficients  are  now  exact  and  the  computation  to  x^  turns  out  to  be  easier 
than  to  x^  by  the  previous  method  ;  the  advantage  introduced  by  the  change  would 
be  even  greater  if  the  expansion  were  to  be  carried  several  terms  farther. 


446  THEORY   OF  FUXCTIOXS 

The  quot'ipnt  of  tiro  power  series  /'(•''')  ^'U  0(:'')j   {f  0{^)  ^^  ^5  may  he 
ohtnined  hij  tJte  ordtnonj  aJgorisni  of  dirisum  os 

f{x)  (1^  4-  f'v''  +   "2-'''  H ,  ,  o     ,  7      ^    A 

For  in  the  first  place  as  y  (0)  ^  0,  the  quotient  is  analytic  in  the  neigh- 
borhood of  ^'  =  0  and  may  be  developed  into  a  power  series.  It  there- 
fore merely  remains  to  show  that  the  coefficients  r,  r^,  r^,  ...  are  those 
that  would  be  obtained  by  division.    Multiply 

and  then  etpaate  coefficients  oi  equal  })0wers  of  .r.    Then 

(',  =  K'o,      "x  =  ^''o  +  ^\fv      "i  =  ^'/o  +  ^\'\  +  ^><r-v  ■  ■  ■ 

is  a  set  of  e(piatioiis  to  be  solved  for  r^^,  r^,  c,.  • .  -.  The  terms  in  /'(•'')  '^^"^ 
(/(■'')  beyond  ./:"  have  no  effect  upon  the  values  of  r^^,  '\:  ■  ■  ■,  ''„,  ^i^d  hence 
these  Avould  l)e  the  same  if  /'„  +  i,  ^',,  +  2;  ■  ■ '  "\vere  replaced  by  0,  0,  •  ■ .,  and 
"„  + 1 ,  ^'„  ^ 2 ;  •  •  •  J  '^'■1  uj  "2n  +  ij  ■  ■  ■  ^T  s^^<'li-  values  (t '„ _^  1 ,  c/ '„  +  .,.  • .  • ,  o '., ^^ ,  0,  .  •  • 
as  would  make  the  division  come  out  even ;  the  coefficients  e  ,  i-,---  r 
are  therefore  precisely  those  obtained  in  dividing  tlie  series. 
If  //  is  developed  into  a  power  series  in  ./•  as 

//  =/(■'•)  =  "„  +  "r'-  +  "./'  +  •  •  •,         ",  ^  0,  (17) 

then  X  may  be  developed  into  a  power  sei'ies  in  //  —  o^^^  as 

X  =/-'(.'/  -  o,)  =  \(y  -  oj  +  hjf/-  n^f  -f  . . ..  (18) 

For  since  c^  ^  0,  the  function /(.r)  has  a  nonvanishing  derivative  for 
X  =  0  and  hence  the  inverse  function/'"^  ( y  —  nj  is  analytic  near  x  —  0 
or  y  —  (1^  and  can  be  develo])ed  (j).  477).  The  nu'tho^l  of  undetermined 
coefficients  may  be  used  to  find  fi^,  ^/„.---.  This  ])r()cess  of  finding 
(18)  from  (17j  is  called  the  rercrslon  of  (17j.  For  the  actual  work  it  is 
simpler  to  re})lace  (y  —  "^)/"^  by  f  so  that 

and        ./•  =  /'  +  I,'/-  +  1,'f  +  h\f''  H ,  ],'■  =  h.n\ . 

Let  the  assumed  value  of  ./•  be  substituted  in  the  series  for  t\  rearrange 
the  terms  according  to  powers  of  /  and  equate  the  corresponding  coef- 
ficients.   Thus  .  ,  ,     , 

+  (a;  +  2 i>'^o:^  +  iKfu'^  +  ■.) Uju  j^n\)t^ ^... 

or     /y.'  =  — •  (•/„,  }i\  =  2  "o"  —  'Vp,  h'  =  —  T)  11',;''  -\-  ~)  ii'ji'„  —  *■/'    . .  -. 


INFINITE   SEKIE.S  447 

170.  For  some  few  purposes,  which  are  tolerably  important,  a  formal 
opirndlonal  viethod  of  treating  series  is  so  useful  as  to  be  almost  indis- 
pensable.   If  the  series  be  taken  in  the  form 

with  the  factorials  which  occur  in  ]Maclaurin's  development  and  with 
unity  as  the  initial  term,  the  series  may  be  written  as 

e-  ^  1  +  n\r  +  ^  ,■-  +  1^  a^^  +  . .  .  +  ^  .r»  +  .  . . , 

provided  that  r/'  be  interpreted  as  the  formal  equivalent  of  *-',•.    The 
product  of  two  series  would  then  formally  suggest 

^ax^bx  ^  ^(a  +  b)x  =  1  +  (,,  _i_  /,y^r.  +  —  (a  +  ///-.,•-  H ,  (19) 

and  if  the  coefficients  be  transformed  l)y  setting  a'h'  —  a/tj,  then 

( 1  +  a^  +  i:  •'^'  +  •  •  •)  (i  +  ^1'^  +  h  ■'■"  +  •  •  •) 

=  1  +  (",  +  />,).'■  +  -^^/:^^^-  ^^^  +  ■  ■  ■  • 

This  as  a  matter  of  fact  is  the  formula  fqv.  the  product  of  two  series 
and  hence  justifies  the  suggestion  contained  in  (19). 
For  example  suppose  that  the  development  of 

X  ,  n.,    ,      B.,    „ 

r    —  1  -  .  o  . 

were  desired.    As  the  development  begins  with  1,  tlie  formal  method 
may  be  applied  and  the  result  is  found  to  be 

'-,  a-  =  e<^^'  +  i^"-t'^%  (20) 


e^  — 1 


X  =  X  +  [( B  +  1  )—  B'^  ^77  +  [<  ^^'  +  1 )'  -  ^"]  ^  +  ■  •  • '         (-1) 

(B  +  i)--B-  =  o,     (/3  +  i)'-^'  =  o,  ••■,     r/?  +  i/"-i:;''  =  o,  ••■, 

(n'  2B^  +  l  =  0,     3  /i,  +  P>  /i,  +  1  =  0,      4  /|^  +  (i  /i,  +  4  7J^  +  1  =  0,  •  •  • , 

or  B^  =  -h,  B.^  =  }„         i?3  =  0,  ii,  =  -3L.... 

The  formal  method  leads  to  a  set  of  equations  from  which  the  suc- 
cessive B's  iiii^y  quickly  be  determined.    Note  that 

X  .'■       .'■  e'  +  1       ./■        -    ./■  .'•         .    /      x\  ,.^... 

7^  +  2  =  2  ^-^i  =  .  -^^^^  2  =  -  2  ""'^^  (~  2)  ^"^ 


448  THEORY  OF  FUNCTIOXS 

is  an  even  function  of  x,  and  that  consequently  all  the  5's  with  odd 
indices  except  B^  are  zero.  This  will  facilitate  the  calculation.  The 
first  eight  even  B'^  are  respectively 

I  1  1_  1„  5  ^fi  9  1  7  3_fi  1  7  (''>?,\ 

ffJ  30'         42!  ;iO'        ^(T)  2  7  .i  0  '         ^^  510    •  \- ^ ) 

The  numbers  B,  or  their  absolute  values,  are  called  tlie  BernouUinn 
numhers.  An  independent  justification  for  the  method  of  formal  cal- 
culation may  readily  be  given.  For  observe  that  e^e'^''  =  ('(■'i  +  '^)^  of  (20) 
is  true  when  B  is  regarded  as  an  independent  variable.  Hence  if  this 
identity  be  arranged  according  to  powers  of  B,  the  coefficient  of  each 
power  must  vanish.  It  will  therefore  not  disturb  the  identity  if  any 
numbers  whatsoever  are  substituted  for  B^,  B'-,  B^,  •  •  • ;  the  particular 
set  B^,  JB.„  iJg,  ••  •  may  therefore  be  substituted ;  the  sci-ies  may  be  rear- 
ranged according  to  powers  of  .r,  and  the  coefficients  of  like  powers  of 
X  may  be  equated  to  0,  —  as  in  (21)  to  get  the  desired  equations. 
If  an  infinite  series  be  written  without  tlie  factorials  as 

1  +  ^'.■'-  +  "/■'  +"3^'  +  •  •  •  +  "„■'■"  H , 

a  possible  symbolic  expression  for  the  series  is 

— =  1  +  (i^x  +  tt'x-  +  cV  -{-■■■,  //'  =  a^. 

1  —  (IX 

If  the  substitution  y  =  •'"/(!  +  ■'')  ov  x  =  ///(I  —  //)  be  made, 

1  1  -  // 


l-(l  +  ")y  (24) 

1-// 

Now  if  the  left-hand  and  light-hand  expressions  l)e  expanded  and  n  ])c 
regarded  as  an  independent  varia])le  restricted  to  values  Avhich  make 
\rix\  <  1,  the  series  obtained  Avill  both  converge  absolutely  and  may  be 
arranged  according  to  powers  of  ".  Corresponding  coefficients  will  then 
be  equal  and  the  identity  will  therefore  not  V)e  disturbed  if  </,  replaces 
a'.    Hence 

1  +  ",■'•  +  f'.'-  -f  •  •  •  =  (1  -  //)  [!  +  (!  +  ")u  +  (1  +  "fif  +•••], 

provided  that  both  series  converge  aljsolutely  for  -'',  =  <i\    Tlien 

1  +  a^x  +  o^r  -f  (/^./■^'  H =  1  4-  ,,y  4-  „  (1  +  „j  f  _|_  ,,  (1  4.  nfif  -\ 

or  (i^C  -f  <i,/-  -f  11./^  -f  . . .  =  r/^y  +  ( ,/j  +  ,/j //- 

+  "'i  +  2.^  +  .g/  +  ....  (25) 


INFINITE  SERIES  449 

This  transformation  is  known  as  Eulei^s  tro.nsfonnatlon.  Its  great 
advantage  for  computation  lies  in  the  fact  that  sometimes  the  second 
series  converges  much  more  rapidly  than  the  first.  This  is  especially 
true  when  the  coefficients  of  the  first  series  are  such  as  to  make  the 
coefficients  in  the  new  series  small.    Thus  from  (25) 

log  {l^o-)  =  x-\  ;/•-  +  ^  a:^  -  i  ,■'  +ix^-\:,-^^... 

To  compute  log  2  to  three  decimals  from  the  first  series  would  require 
several  himdred  terms  ;  eight  terms  are  enough  with  the  second  series. 
An  additional  advantage  of  the  new  series  is  that  it  may  continue  to 
converge  after  the  original  series  has  ceased  to  converge.  In  this  case 
the  two  series  can  hardly  be  said  to  be  equal ;  but  the  second  series  of 
course  remains  equal  to  the  (continuation  of  the)  function  defined  by 
the  first.  Thus  log  3  may  be  computed  to  three  decimals  with  about  a 
dozen  terms  of  the  second  series,  but  cannot  be  computed  from  the  first. 

EXERCISES 

1.  By  the  multiplication  of  series  prove  the  following  relations: 

(a)  (1  +  a;  +  x2  +  x3  +  .  •  •)2  =  (1  +  2x  +  3x2  +  4x3  +  . . .)  =  (1  _  ^yi^ 

(p)  cos2 X  +  sin2 X  =  1,         (7)  e^c'J  =  e^+  ",         (5)  2  sin^ x  =  1  —  cos 2 x. 

2.  Find  the  Maclaurin  development  to  terms  in  x^  for  the  functions: 

{a)  e^cosx,         ((3)  e^siux,         (7)  (1  +  x)log(l  +  x),         (5)  cosxsin-ix. 

3.  Group  the  terms  of  the  expansion  of  cosx  in  two  different  ways  to  show  that 
cos  1  >  0  and  cos  2  <  0.    Why  does  it  then  follow  that  cos  |  =  0  where  1  <  ^  <  2  ? 

4.  Establish  the  develoi:)ments  (I'eirce's  Xos.  785-789)  of  the  functions: 

(a)  e^'"-^,         (/3)  t"^"-'',         (7)  e«'"~^^,         (5)  gtan-i^. 

5.  Show  that  if  g{x)  =  h,„x">  +  6,„  +  ix'"  +1  +  •  •  •  and/(0)  ^i  0,  then 

f(x)        n,.  +  a,x  +  a.^x- +  ■  ■  ■        c_,„      f-m  +  i  c_i 

f/(x)      6„.x"' +  6,„  +  ix'«  +  i  +  ■  ••        X'"         x"'-i  x 

and  the  development  of  the  ciuotient  has  negative  powers  of  x. 

6.  Develop  to  terms  in  x^  the  following  functions: 

{a)  sin(tsinx),         (/3)  logcosx,         (7)  Vcosx,         (5)   (1  —  ^-^  sin^x)"  2. 

7.  Carry  the  reversion  of  these  series  to  terms  in  the  fifth  power: 

(a)  \i  —  siiix  =  X  —  4  X-'  +  •  •  ■ ,  {(3)   //  =  tan-i  X  =  x  —  |-  x^  +  •  ■  • , 

(7)  ?/  =  e^.-  =  1  4.  X  +  I  X-  +  •  •  • ,         (5)  ,v  =  2  X  +  3 x2  +  4 x3  +  5  x'*  +  ■  ■  • . 


450  THEORY  OF  FUNCTIONS 

8.  Find  the  smallest  root  of  these  series  by  the  method  of  reversion: 

(a)  -  =  f'c-Alx  =  x--x^+  ~-,x^  -  — ^x^  +  • .  ■, 
2      ^0  o  .5.0  6  .  i 

(^)  \  =  (\o^xMx.         (7)  -  =  f  " 

4      Jo  10      Jo    V(l- j-')(l-  ix^) 

9.  By  the  formal  method  obtain  the  general  ecinations  for  the  coefficients  in  the 
developments  of  these  functions  and  compute  the  first  five  that  do  not  vanish: 

(«^)    -^ 7'  (A)    — -T'  W 


t"'-  —  1  e'  +  1  1  —  2  xe-''  +  e-^ 

10.  Dbtain  the  general  expressions  for  the  following  developments: 

1       .r       x'^       2x''  li:,A-2x)-" 

(a)  coth x  =  -H -\ +  -= , 

^    '  X      S      45      945  (-Ihj'.x 

1      X      j3       2.r^  ,   ,      ^^    7)'o„(2j-)2« 

(/3)  cot x  = +  (-1)"  — — -— . 

^   '  x      3      45      1145  '      {iitj'.x 

,    ,   ,        .  ,  a-"        x^  j-«  ,   /      •,      B:„{2x)-" 

(y)  log  sin  X  =  logx +  (-  1)" , 

x-       x^  x'5  7>o„(2x)-« 

(  5 )  log sndi  X  =  logx  -\ 1 •  •  •  4 = •  •  • . 

^    '      -  '-  c,        180      28::!5  2  )i  •  (2  n) ! 

11.  The  Eulerian  numbei's  E->„  are  the  cnefficients  in  the  expanslDU  of  sech  x. 
l--srablish  the  defining  equations  and  t'onipute  the  first  four  as  —  1.  5.  —  01.  1385. 

12.  Write  the  expansions  for  sec  x  and  log  tan  (i  tt  4-  ^  x). 

1  2  1 

13.  From  the  identitv =  derive  the  expansions: 

'   t-'-  -  ]       e-'  -  1       t"'-  4-  1 

(a)  ^f^  =  \  +  13,(2^  -  1)  "-;  4-  «,(2^  -  1)  ^  +  •  •  •  4-  7J.„(2-''  -  1)'^'  4-  •  •  • , 
e'  4-  1       2  2  I  4  ;  2  ?;  ! 


+ 

11  r  r^  (.l!/)-! 

^  7J^(,.  _  1,       _  «^(24  _  1 )  _ /i,„(,.»  -1)--^-^ 

4-12  2  1  4  .  2 )(  : 

X- "  ■ 


( 7 )  taidi  X  =  (2-  -  1 )  2-B.,  ~  4-  (2^  -  1 )  2-*  />'  --    4-  ■  •  •  4-  (2-  "-1)2-  "JL  „ ' +  •  •  ■ , 

'21  41  2/(1 

(5)  tan  X  =x  +  ~  +  ~^  +  -^^  +  ■■■  +  (-  ^Y'-HS-"  -  1)  2- " /i.  „  "'^^^^  4-  ■  •  • . 
■J         ]•)         ol-)  2  )i  1 

(  e  )  loi:r  cos  X  =  —  ■ ' ''    _..._(_  l)"  -i(2-"  —  1 )  2■-"7^„    -    — •  •  •  . 

2        12       45  ■    2)/  •2h  1 

(f)  log  tanx  =  loiTX  4-  "' -  +  -''^-  +  ...  +  (-  ly, -i{-s->- --l  _  ] ,  i'-.; » /i.,  „  _Zl- _  4-  ■  •  • . 
•"]         (i()  )(  ■  2  ?i  1 

(77)  cscx  =  -  (cot"-^  4-  tan"'')  =-+-  +  ...  +  (_  ij^i-i  2  (2-''-i  -  1)7I,„  '''—, 
2  \        2  2  /       X       ;!  1  2 »  1 

iff)   lo-ccishx.  (i)   logtanhx.  (k)   cS(4i  x,  (\)   sec-x. 


INFINITE   SEFvIES  451 

Observe  that  the  Bernoullian  miinbers  afford  a  general  develnpinent  for  all  the 
triuonoinetric  and  liyperl)olic  functions  and  tlieir  ioiiarithms  with  the  exception  of 
tlie  sine  and  cosine  (wliich  have  known  devehipnients)  and  the  secant  (wliich  re- 
ijuires  the  Eulerian  numbers).  The  importance  of  these  numbers  is  therefore 
ajiparent. 

14.  The  coefficients  P^iy),  P.i{{/),  •  ■  ■ .  P„{y)  in  the  development 


e^-  1 


y  +  P^x  +  P.,{y)-^-  +  ■  •  •  +  Pn{y)-t>'  + 


are  called  Bernoulli's  polynomials.   Show  that  {n  +  1)  !  Puiy)  =  {B  +  y)"  +i  —  7i''+i 
and  thus  compute  the  fir.st  six  polynomials  in  y. 

15.  li  y  =  X  is  a  positive  integer,  the  (juotient  in  Ex.  14  is  simple.    Hence 

n  :  7^,(.V)  =  1  +  2«  +  3"  +  •  •  •  +  {X-  1)« 

is  easily  shown.    With  the  aid  of  the  polynomials  found  above  comimte: 

(a)   1  +  2^  +  ?>^  +  •  ■  •  +  10^  (^)   1  +  2'  +  3-'  +  •  •  ■  +  ;»■■. 

(7)1  +  2--^  +  .3-^  +  .  .  .  +  {X  -  1)-,         (5)   1  +  2^'  +  3-'  +  .  .  .  +  (.Y  -  1  )3. 

_  '  r  1  _  l>,  +  1 

■  (a  -  b)  I  I  -   >  — 


16.  Interpret = =   >  

1  —  ax  1  —  bx      X  [a  —  b)  l_l  —  ux      a  —  bxj       *—l        n  ■ 


17.  From  I     e-fi-"-'.)'(J^  = establish  formal! v 

•J»  \  —  ax 

1  +  o,J  +  "..•'•-  +  (I  -x^  +  •  •  •  =  r    t-  'F{xt)dt  =  -   r    e~^  F(u)du. 

where  F{h)  =  1  +  «,h  -| «.,»-  -\ a.,H^  +  •  •  •• 

2  1"  3  !    '^ 

Show  that  the  integral  will  converge  when  0  <  /  <  1  provided  j«,j  ^  1. 

18.  If  in  a  series  the  coefficients  a;  =  /    i'f(t)<U.  show 

1  +  «i.f  +  «.,J--  +  <i.iX^  +  •••=/     - 

t/  0    1 


^'''  at. 


It 

19.  Note  that  Exs.  17  and  18  convert  a  series  into  an  integral.    Show 
^    ^  2/'       3p       4/'  r(]>)Jo         I-./-/  ^         ),/'        Jm 


'l  +  l-       1  +  2-'       l  +  3--^  Jo      1-J<  '     l  +  7r       .'o 


1        ,        /  .r-  /^  1  sin  log^  ,^     ,  1 

T2^'  "^  lT3-  "^  "  '  ~  ~  Jo    "l 

«(«  +  ])  „.,   ,    "(f'  +  l){a  +  2)  ^3 


(7)   1  +  -  X  +  ^ — ■ — '-  X-  + —      "  ./•■'5  + . 

^^  b  h{b->r\)  6(6 +  !)(/> +  2) 


r(a)r(6-a)  Jo 


(a)r(6-a)Jo  1-j-i 


452  THEORY  OF  FUNCTIONS 

20.  In  case  the  coefficients  in  a  series  are  alternately  positive  and  negative  show 
that  Euler"s  transformed  series  may  be  written 

a^x  —  02^2  +  asx^  —  ntx*  +  •  ■  •  =  a^ij  +  Auiy-  +  A^ait/^  +  A^aiy*  +  ■  ■  ■ 

where   Aai  =  Ui  —  ao,    A'-«i  =  Ac(i  —  Ado  =  (ii  —  2  a-i  +  t/g,  ■  •  •    are   the   successive 
first,  second,  •  •  •  differences  of  the  numerical  coefficients. 

21.  Compute  the  values  of  these  series  by  the  method  of  Ex.  20  with  x  =:\.y  —  I. 
Add  the  first  few  terms  and  apply  the  method  of  differences  to  the  next  few  as 
Indicated : 

(a)  1 \- f----=  0.69315.  add  8  terms  and  take  7  more, 

^    '  2      8      4 

{(i)   1 1 +  •  .  ■  =  0.1)040,  add  5  terms  and  take  7  mure, 

V^       \''3       V4 

(y)  '^  =  i_^-|-l_^  +  ...=  0.78539813.         add  10  and  take  11  more, 
4  3      5      7 

/         111  \  2/'-i      /  111 

(5)   Prove    1+  -  +  -  +  —  +  ...=  1 + +  . 

^    '  \        •>!'       3/'       4/'  /       2/'  - 1  —  1  \        -li'      3/'       4/' 

and  compute  iov  p  =  1.01  with  the  aid  of  five-place  tables. 

22.  If  an  infinite  series  converges  absolutely,  show  that  any  infinite  series  the 
terms  of  which  are  selected  from  the  terms  of  the  given  series  nuist  alscj  cunverge. 
What  if  the  given  series  converged,  but  not  absolutely  ? 

23.  Note  that  the  proof  concerinng  term-by-terin  integration  (p.  432)  would  not 
hold  if  the  interval  were  infinite.  Discuss  this  case  with  especial  rt-ftrences  to 
justifying  if  possible  the  formal  evaluations  of  Exs.  12  (cr),  (5),  p.  439. 

24.  Check  tlie  formula  of  Ex.  17  by  termwisc  integration.    Evaliu\te 

^   f     e   ■'■JAhu)  du  =  1  -  ',  h-x-  +  i  •  ^ =  (1  +  '^-■c-)~  2 

X  Jii  '  '"21 

by  the  invt-rse  trausformatii.iu.    Set'  Exs.  8  and  15.  p.  3'.i9. 


CHAPTER   XVII 

SPECIAL  INFINITE  DEVELOPMENTS 

171.  The   trigonometric   functions.     If  m   is  an  odd   integer,   say 
///  =  li  «  +  1,  ])e  Moivre's  Tlieoreiii  (§  ~2)  gives 

sin  y//(i  ,  (m—l)(in  —  2)         ,      ,       .    , 

^^  =  cos-"  <i>  - '~f, ^  (•()s-"--ci  sm-  <i  H ,        (1) 

in  sm  (^  .^  I 

where  by  virtue  of  the  relation  cos'- (^  =  1  —  siir-<^  the  right-hand  mem- 
ber is  a  polynomial  of  degree  n  in  sin'-  <^.  From  the  left-hand  side  it  is 
seen  that  the  value  of  the  polynomial  is  1  when  sin  eft  =  0  and  that  the 
ii  roots  of  the  polynomials  are 

sin'-  tt////,  sin'-  2  7^///^       '    •  •  • ,  sin'-  nir J m. 

Henct^  tlip  ])olynomial  niay  l)e  factored  in  the  I'orm 

sin  m<^  =i\-     ^"^'^   \  A ^"''<^     \  . . .  A  _      '^"^'  ^     \       /ox 


/// sin  <^       Y         sin'-TT//// /  \         ^m-'lTr/nij        \         Hu\-7i7r/i/i 

If  tlie  substitutions  ^  =  .r/i/i  and  ^  =  ix/m  be  made, 

sin  .'/•  / _,        sin'- ./■//// \ /^         sin^.r/'"  \         /^         sin^  »:•//»  \      „ 

m am  x/iii.       \  am-  tt/iii  /  \         sur  J  tt////  /         \         aurinr/inj     ~  ^ 

_jdidKr^^/    _^sin^ 

//^.sinhay/M       y  sur7r/m/\         hiu-'Jtt/j// /         \         surynr/m/    '    ' 

Now  if  ill  be  allowed  to  l)ecome  infinite,  passing  tlirough  successive 
odd  integers,  these  equations  renuiin  true  and  it  would  apjiear  that  the 
limiting  relations  would  hold  : 


J'  \  it'-  j  \  'I'  TT'  J  I    \  IriT- 

siidi  X 


,1  x"" 
snr-  (-_  +. 

()  in 


hm 


A'TT        ,„=x  //.'TT        1  I  L'ttY  \'-        A''-7r'" 

~  ( 
453 


snr—      ■-- r.    —     + 

7/i  \  //^        0 


454  THEORY   OF   FUNCTIONS 

In  tliis  wuy  tlte  ej'jjfnuion.s  Into  injinite  products 

sin  x  =  ./•  TT  ( 1  —  ~ — ,  I  -•         sink  x  =  »■  TT  ( 1  +  t^— ,  |  (o) 

1   \         IriT-j  1   \         k-ir-j 

would  be  found.  As  the  theorem  that  the  limit  of  ii  })r()duct  is  the  prod- 
uct of  the  limits  holds  in  general  only  for  finite  products,  the  process 
here  followed  must  be  justified  in  detail. 

For  the  justiticatioii  the  consideration  of  .'<inli  j,  which  involves  only  positive 
quantities,  is  simpler.    Take  the  logarithm  and  split  the  sum  into  two  parts 


sndi--  „  /         sndi 


Off =  >  loi;-    1  + +  >  loir    1  -j- . 

?HSudi—        1  \  sm- —  /      V' - 1        \  sin-  —  / 

rti  ^.  //(  ?/i  / 

As  log(l  +  a)  <  (t.  the  second  sum  may  be  further  transformed  to 

>i  /         sinh2  — \         „     sinh"^^  „ 

Jt'  =  >    loff     1  + <>    =  sndi^-> 

j^+i  \  sm^  —  /      J'  +  i  sm- —  ^'-ism-  — 

\  HI '  m  in 

Now  as  ?i  <  1  »i,  the  angle  kir/m  is  less  than  \  ir.  and  sin  ^  >  2  ^/tt  for  ^  <  i  tt,  by 
Ex.  28,  p.  11.    Hence 


R  <  sudi-  -    > =  — smli2—   X  — ;  < — snih^—   \ 

in  -^  4  k-        -1  //(  -^^,  k-        4  m  Jp 

P  ~   i.  /'  T   1 


1        m-   .    ,  „  X    c'^-  dk 


n     X 


j,     /         siidi- 
1         sinh  ,r         xr^  I  -,    ,              "'1       '"^    •   i  ,  •'^ 
Hence  log >      1  -\ <        smh^— . 

•   ,   •'■       ■^  •    J'-""'         -^P  "1 

m  sinli    -        1     \  sni-  —  / 

Now  let  m  become  intinite.    As  the  sum  on  the  left  is  a  finite,  the  limit  is  simply 

sinh  J-      -^^A  /■           .r-  \       j--           ,  ,      sinh  j-      x^  /,         /-  \ 
loLT >      1  + <  —  ;  and  loff =  >      ]  + \ 

then  follows  easily  tiy  letting  p  liecome  infinite.    Hence  the  justification  of  (4'). 

])y  tlic  (litl'tM-ciitiatiun  of  the  series  of  logarithms  of  (o), 

sin  ./•       ./^  ,        / ^         ./•-    \  ,       sinh  .r       ^  ,       / ^         ,/■■  \ 

the  expressions  of  cot,'-  and  coth  .r  in  series  of  fractions 

cot  ./•  = y  , .,  ~  ■  —:,,  coth  ,/■  =  -+  y  Tv^H :,  (") 


SPECIAL   INFINITE   DEVELOPMENTS 


455 


are  found.  And  the  differentiation  is  legitimate  if  these  series  eonvcrge 
uniformly.  For  the  hyperljolic  function  the  uniformity  of  the  conver- 
gence follows  from  the  J/-test 

, .,   o   , — r,  <  TT— , '    and     Xtt— ,  converges. 
IriT-  +  ■'■"        A"7r""  ^^  IriT' 

The  accuracy  of  the  series  for  cot  a:  may  then  be  inferred  I)}-  the  substi- 
tution of  /./'  for  ./•  instead  of  bv  direct  examination.    As 


2j' 


Ix'-tt'-  —  ./■'■ 


1  1 


cot  J- 


(8) 


In  this  expansion,  however,  it  is  necessary  still  to  associate  the  terms 
fur  /.•  =  -(-  7t  and  1:  =  —  n  ;  for  each  of  the  series  for  /.•  >  0  and  for 
/.•  <  0  diverges. 

172.   In  the  series  for  cotli.r  rej.lace  x  liy  ],,,■.    Tlien,  by  (22),  p.  447, 


^coth^=i+2 ,7^^^=i+i;^^-'f^ 


(^•) 


If  the  first  series  can  l)e  arranged  according  to  powers  of  x,  an  expres- 
sion for  iJo„  will  be  found.    Consider  the  identity 

if ,  =  - 1  <  -  '>'  -  r+' ' = -  "i  (-  "'■  - "'-  '>"' 

which  is  derived  by  division  and  in  which  6  is  a  })roper  fraction  if  t  is 
positive.    Substitute  ;"  =  x-/\  Irir'- ;  then 


4  IriT-  + ./- 


2 


4  Irir- 


4  ;.•  V- 


-•^^[^)Xw.^-'^^{-li.)X^r 


Let 


X         .    X 

-  coth  - 


^1,1       1 

1  ~  'J 


1  0  s.,, 


"\4  7r- 


*  The  d  is  still  a  proper  fractinu  since  each  Oj^.  is.  The  interchan.ue  <if  tlie  unlcr  of 
suniiiiatioH  is  Icicitiiiiate  l>ecau.se  the  series  would  still  converge  if  all  signs  w  ere  positi\e, 
since  ^k~-''  is  cnn\-frircnt. 


456  THEORY  OF  FU^^CTIONS 

As  .S.,„  approaches  1  wlien  ti  becomes  inlinite,  the  last  term  approaches 
0  if  ^-  <  2  TT,  and  the  identical  expansions  are 

2 1  S,,.(-  ly  - '  -0^.  =  t  ih,.  1^,  =  \  coth  \  -  1.  (10) 

Hence  -B.,.  =(- 1)""' ^1^^?^ «.,  (H) 


and 


|coth|  =  l+i;i?.,^  +  ^i3.„:J^-        (12) 

w  ^  "^  /'    •  -ill. 

The  desired  expression  for  B.,^^  is  thus  found,  and  it  is  further  seen 
tliat  the  expansion  for  \  x  coth  \  x  can  be  broken  off  at  any  term  with 
an  error  less  than  the  first  term  omitted.  This  did  not  appear  from  the 
formal  work  of  §  170.  Further  it  may  be  noted  that  for  large  values  of 
n  the  numbers  B.^^  are  very  large. 

It  was  seen  in  treating  the  T-f unction  that  (Ex.  17,  p.  385) 

log  r(?i)  =  {n  —  \)  log  71  —  n  +  log  V2  TT  +  o)  (?i), 


■^"^^Ki 


where  w  (n)  =  /      (  -  coth  -  —  " 


As  I      x-''t''"dx  =  I      .T-^'e    ^^dx  = 


_r(2;j  +  l)  _    2^^! 


the  substitution  of  (12),  and  the  integration  gives  the  result 

-('^)  =  T:2-  +  Tnr  +  ---  +  (2,.-a)(2,.-2)  +  (2,.-i)2/(i^) 

For  large  values  of  n  this  develo})ment  starts  to  converge  very  ra])id]y, 
and  by  taking  a  few  terms  a  very  good  value  of  w(«)  can  be  obtained  ; 
but  too  many  terms  must  not  be  taken.    Compare  §§  151,  lo-l. 


EXERCISES 

I     ^  sin  2,/'       _:,  /  4  r- 

1.  Prove  cos  j:  =  ,    .        =  TT  (  1 


ii^i".'-        0    \         (2A:  +  l)-7r- 

2.  On  the  assniuption  that  the  i)rodnct  for  sinh  x  may  be  mnltiplied  out  and 
collected  according  to  powers  of  j:.  show  that 

(a)  V  r;  =  —  .         (/i)  V  V =  ~  ,    wliere  k  ^  I. 


(7)7      -  =       .         (5)7     7    —  =       '     if  A-  may  equal  i 


SPECIAL  INFINITE  DEVELOPMENTS  457 

111  2 

3.  By  aid  of  Ex.  21  (c),  p.  452,  show :    (a)   1  +  —  +  —  +  ~  +  ■  ■  ■  ='^ , 

,       ,        111  7r2  ,,,111  7r2 

(|3    1  + 1 h  —  H =  — .         (7)1 1 1-- 


32      52      72  8  ^"  22      32      42  12 

4.  Prove:     (a)    /     -^-tZx  =  -  ~.        (^)    /     — —  dx  =  -  - , 

Jo    \  —  X  b  .vo     1  +  X  12 

,     ,      /'I    10o;X     ,  TT^  ,,,      /'^         1  +  X  CZX         TT^ 

(7)    /     r^~7A^  =  -^'-       (5  log:, =  -r- 

^ol  —  X-  8  Jo         1  —  XX        4 

5.  From  tan  x  =  —  cot  ( x tt  )  =  —  7 

V         2    /  Ax-(A:  +  -|)7r 

1/       X      ^      x\      ^(-1)^-       1      ^(-l)*-2x 

show  CSC  X  =  -  I  cot  -  +  tan  -    =  >   ^ =     -f  >    -!^ '- 

2  \       2  2/      ^  X-  kir      x      ^  x^  -  k'^TT- 

n-l  ^^  »-l 


(-1)^ 


dx  =  7  -^ ^ ,  and  compute  for  a  =  -  by  Ex.  21,  p.  452. 

u    l  +  x  V  "  +  ^  * 

7.  If  ((  is  a  proper  fraction  so  tliat  1  —  a  is  a  proper  fraction,  show 

Jo    l  +  x       ^a  —  k       ^1    1  +  x  Ji.)     l-\-x  sin  «7r 

1 

8.  When  n  is  Large  B-2n  =  (—  1)"~^  4V7r>i( — )     approximately  (Ex.  13). 

\7re/ 

^  92 

9.  Expand  tlie  terms  of  -  coth  -  =  14-7 by  division  when  x  <  2  tt 

2  2  A'  4  A;27r2  +  x^ 

and  rearrange  according  to  powers  of  x.    Is  it  easy  to  justify  tliis  derivation  of  (11)  ? 
10.  Find  co'(?i)  by  differentiating  under  the  sign  and  substituting.    Hence  get 
r'(n)       ,  1         /i,        B,  B.2,-2  0B.„ 


r  (n)        "^  2n      2n-      4  )i*  (2p  — 2)n2/'-2      2^^)12^^ 

11.  From  — ^—^  +  7=1 da  of  §  149  sliow  that,  if  n  is  integral, 

r  (n)  Jo       1—  a 


(«) 

r'(w)  ,11  1  ,  r'(i)     ^  _„,,, 

— !-i+ 7  =  1 +  -  +  -  +  ...  + ,     and     7  = !^  =  0.57721; 

r(n)  2      3  71-1  r(l) 

by  taking  n  =  10  and  using  the  necessary  number  of  terms  of  Ex.  10. 

12.  Prove  log  T  {n  +  \)  =  n  (log  ?i  —  1)  +  log  V2  tt  +  w^  (n),  where 

pf>  I  1         e"      I       dx 

w  (ti)  =  I      I ;e'"'  — ,         uJn)  =  w{n)  —  w{2n), 

J-jo\x      e^  —  1/         X 

,,      B.,n-^ /^      1\       B,n-^(^       1\      B,n-^ /,       1\ 

w,  (n)  =  -i 1  -  -    +  ~ I  1 I  +  — I  1 + 

^^  '        1  •  2    \        2/        3  .  4    I         23/        5  •  (5    \         2"/ 


458  THEORY   OF  FUNCTIONS 


9  1 


13.  Show  7J  :  =  V^Trn  (- I  e     '    or    V2  tt  ( — —^]      "e    ^*"  +  ^-=.     Note  that  the 


results  of  §  149  are  now  ulitaincfl  rifjorously. 


n-l 


1  ■^-V  fi-».r  .^-^  g— (n— l)x 

14.  From =  >   e-  '■'  -I =  >   e-^^  +  0 ,  and  the  formulas 

1-e-     A  l_c--      ^ 

of  §  140,  prove  the  expansions 


{a) 


f^  log  r  in)  =  V  ^   ,       (^)  f  log  r  (,o  +  7  =  V  (-^  -  -^-\  ^ 


173.  Trigonometric  or  Fourier  series.    If  the  series 

/(•^)  =  i  "o  +2  O'i-  cos  A-.r  +  h„  sill  7.vr) 

=  -J-  c'^  +  ((^  COS  .r  +  i^^,  cos  2  .r  +  (^/^  cos  ^  x  -{-  ■  ■■  ^  ^ 

+  Z*j  sin  X  -\-  h,,  sin  2  x  -{-  b^  sin  3  .r  +  •  •  • 

converges  over  an  interval  of  lengtli  2  tt  in  .r,  say  0  S  .r  <  2  tt  or 
—  TT  <  .'■  ^  TT,  the  series  Avill  converge  for  all  values  of  x  and  Avill  de- 
fine a  periodic  function  y(.'/;'  +  2  tt)  =  /(■'')  of  period  2  tt.    As 


(       cos  /.•,?  sni  Z,rc/,/'  =  0     and       |  .     ^       .     ,    ax  =  ()  or  tt     (1 

Jo  J,.       s"^  ^-^  sm  ^.^• 


'^) 


accoi'diug  as  /.•  ^  ^  or  /.•  =  I,  the  coefficients  in  (14)  may  be  determined 
formally  by  multiplying /"(^.r)  and  the  series  by 

1  =  cos  0  X,  cos  X,  sin  x,  cos  2  .r,  sin  2  x,  ■  ■  ■ 

successively  and  integrating  from  0  to  2  tt.     By  virtue  of  (15)  each  of 
the  integrals  vanishes  except  one,  and  from  that  one 

1    r-''  1    r-'^ 

n,.  =  -     /       fix)  COS  Ixdx,  h,.  =  -     I       f(.r)  sin  Lnlx.        (10) 

Conversely  if  _/'(./■)  be  a  function  which  is  defined  in  an  interval  of 
length  2  77,  aiul  which  is  continuous  except  at  a  finite  numl»er  of  ])oints 
in  the  interval,  the  numl)ers  "^.  and  h^.  may  be  computed  according  to 
(H))  and  the  series  (14)  may  then  be  constructed.  If  this  series  con- 
verges to  the  value  of /'(•'•),  there  has  been  found  an  ex])ansion  of /(,r) 
over  the  interval  from  0  to  2  tt  in  a  fn'f/nnomefri<'  or  Fmiricr  scries* 
The  question  of  whether  the  series  thus  found  does  really  converge  to 

*  ISy  special  devices  some  Fourier  expansions  were  found  iu  Ex.  10,  p.  4o9.   . 


SPECIAL  INFINITE  DEVELOPMENTS  459 

the  value  of  the  function,  and  whether  that  series  can  be  integrated  or 
differentiated  term  by  term  to  find  the  integral  or  derivative  of  the 
function  will  be  left  for  special  investigation.  At  present  it  will  be 
assumed  that  the  function  may  be  represented  by  the  series,  that  the 
series  may  be  integrated,  and  that  it  may  be  differentiated  if  the  differ- 
entiated series  converges. 

For  example  let  &  be  developed  in  the  interval  from  0  to  2  tt.    Here 
a/.-  =  -  I      e^  cos  Jucdx  =  —  ^  cos  ydy  =    ^   -| — ^ ) 

TTt/O  KTJ-  J  0  L7r\A:''  +  l         /Jo 

1  „      1  1  „      1        11 

or  a    =  —  e-  "■ ?        cik  =  -"-'^ 


^^  +  1        TT  k-  +  1 

1    /'-''•,     7  1    o         ^  1       fc 

and  bh  =  -   I       e-^  sni  kjcdx  = e-'^ i 

IT  Jo  TT         ^2  +  1       7rA:2  +  i 

7re-^  1  1  1  o  1  o 

Hence  =  -  H cos  x  -\ cos  2x  -\ cos  3  x  + 

e2  TT  _  1      2      12  4-  1  22  +  1  32  +  1 

12  3 

sni  2  X sin  3  x  +  • 


12  +  1  22  +  1  32  +  1 

This  expansion  is  valid  only  in  the  interval  from  0  to  2  tt  ;  outside  that  interval  the 
series  automatically  repeats  that  portion  of  tlie  function  which  lies  in  the  interval. 
It  may  be  remarked  that  the  expansion  does  not  hold  for  0  or  2  tt  but  gives  the 
point  midway  in  the  break.  Note  further  that  if  the  series  were  differentiated  the 
coefficient  of  the  cosine  terms  would  be  1  +  1/A;2  and  would  not  approach  0  when 
k  became  infinite,  so  that  the  series  would  apparently  oscillate.  Integration  from 
0  to  X  would  give 

7r(e''-l)       1  1        .  1        sin2x  1       sin3x 

— ^: '-  =  -  X  -\ Sin  X  H 1 1-  •  ■  • 

e2,r_i        2         12  +  1  22  +  1       2  32  +  1       3 

-\ cos  X  H cos  2x4 cos  3 x  +  •  •  • , 

12  +  1  22  +  1  .32  +  1 

and  the  term  ^x  may  he  replaced  by  its  Fourier  series  if  desired. 

As  the  relations  (15)  hold  not  only  when  the  integration  is  from  0 
to  2  TT  but  also  when  it  is  over  any  interval  of  2  tt  from  a  to  a  +  2  tt, 
the  function  may  be  expanded  into  series  in  the  interval  from  a  to 
«:  -!-  2  TT  by  using  these  values  instead  of  0  and  2  tt  as  limits  in  the 
formulas  (16)  for  the  coefficients.  It  may  be  shown  that  a  function 
may  be  expanded  in  only  one  way  into  a  trigonometric  series  (14)  valid 
for  an  interval  of  length  2  tt  ;  but  the  proof  is  somewhat  intricate  and 
Avill  not  l_)e  given  here.  If.  however,  the  expansion  of  the  function  is 
desired  for  an  interval  a  <  x  <  (3  less  than  2  tt,  there  are  an  infinite 
number  of  developments  (14)  which  will  answer ;    for  if  <fi  (.r)  be  a 


460  THEORY  OF  FUXCTIOXS 

function  Avhich  coincides  with  /(•^■)  during  the  interval  a  <  x  <.  (i, 
over  which  the  expansion  of /(./•)  is  desired,  and  wliich  lias  any  value 
whatsoever  over  the  remainder  of  the  interval  /3  <  .r  <  «:  +  2  tt,  the 
expansion  of  <^  (x)  from  a  to  a  -\-  2  tt  will  converge  to  /(.'')  over  the 
interval  a  <.  x  <C  /3. 

In  practice  it  is  frequently  desirable  to  restrict  the  interval  over 
which  f(x^  is  expanded  to  a  length  tt,  say  from  0  to  tt,  and  to  seek  an 
expansion  in  terms  of  sines  or  cosines  alone.  Thus  suppose  that  in  the 
interval  0  <  a;  <  tt  the  function  <^  (.'■)  be  identical  with  f(x),  and  that 
in  the  interval  —  tt  <  ,r  <  0  it  be  equal  to  /(—  x)  ;  that  is,  the  func- 
tion (f>(.r')  is  an  even  function,  </>(■'•)  =  </>(—.'■),  which  is  equal  to /(.<■) 
in  the  interval  from  0  to  tt.    Then     , 

({)  (x)  cos  kxdx  =  -    j     <f>  (^')  cos  kxdx  =  2    /      f  (,/•)  cos  Jcxdx, 

IT  Jl)  Jf) 

cf>  (,/•)  sin  L'xdr  =   /     (^  (x)  sin  I'xdx  —  I     cf)  (,r)  sin  J:xdx  =  0. 

Hence  for  the  expansion  of  </>(•'■)  from  —  tt  to  +  tt  the  coefficients  /i^.  all 
vanish  and  the  exjiansion  is  in  terms  of  cosines  alone.  As  /(.'')  coin- 
cides with  (f)  (x)  from  0  to  tt,  the  expansion 


r> 


/(:'-)  =X"^^'^^^'''''  "'^■^i    /     /(•'■)  cos  A-a;^/^  (1.) 

0  Jl) 

of /(.r)  in  terms  of  cosines  alone,  and  valid  over  the  interval  from  0  to 
TT,  has  been  found.    In  like  manner  the  expansion 

/(■'■)=yf>,smLr,  /.,  =  -    r  /(,r)  sin  A-.Tr/.r  (18) 

1  '^  Jl) 

in  term  of  sines  alone  niay  be  found  by  taking  (pix)  etpial  tof{x')  fi'om 

0  to  TT  and  equal  to  —  ,/'(^  ■'')  f^'oi^^  0  to  —  tt. 

Let  |x  be  developed  into  a  .series  of  siiies'and  into  a  .series  of  cosines  valid  over 
the  interval  from  0  to  tt.    For  the  .series  of  sines 

2    /"^  1       .     ,     ,  (—1)^'  .f      A       sinfcx 

III-  =       I      -  .-f.  sni  kxdx  —  —  -^^ ,         -  =y    ± 

TV  Jo     2  k  2      ^  k 

or  I  X  =  sin  x  —  i  sin  2 a-  -f  J  sin  3x  —  ^  sin  4 .c  +  •  •  • .  (A) 

„  ,  ,  ,  r  *^.  k  even 

Also       (in  =  ~    I      -./■'/.(■=-,         (1^.  =        I      -  X  coakxdx  =  J,         2 

TT  Jo     2  2  TT  J<j     2  I :  k  odd. 

I       •^'^■ 

..  I  TT        2  I  ros;i,c        cos")./-        rosT./'  1  ,„, 

Hence  -x  .^      -        cis x  +  -  —^  +  -  -_   -  +  —^---  +  •  •  •    ■  (B) 

2  4       77  .3-  o-  ( - 


SPECIAL   INFINITE  DEVELOPMENTS 


4(31 


Although  the  two  expansions  define  the  same  function  ^  x  over  tlie  interval  0  to  tt, 
they  will  define  different  functions  in  the  interval  0  to  —  rr,  as  in  the  figure. 
The  development  for  i  x-  may  be  had  by  integrating  either  series  (A)  or  (B). 


Ix"  =  1  —  cos  X  —  i  (1  —  cos  2  x)  +  1(1—  cos  3  x)  —  j'g  (1  —  cos  4  x)  + 

TT         2  r  .  sin  3x      cos5x  1 

—  —  X sin  X  H 1 !-•••• 

4  ttL  33      ^      53  J 


33  53 

These  are  not  yet  Fourier  series  because  of  the  terms  I  ttx  and  the  various  l\s.  For 
^TTX  its  sine  series  may  be  substituted  and  the  terms  1  —  ^  +  ^  —  •  •  •  may  be  col- 
lected by  Ex.  3,  p.  4.57.    Hence 


(-tt.tt) 


1    o      ■n-2  1         ,         1        o  1         . 

-  X-  = cos  X  +  -  COS  2  X cos  3  X  +     -  cos  4  x  — 

4  12  4  9  16 


(A') 


1    ^        2r/7r2        A     .  TT-    .     ,  (it-        1\    .     o  -^'^    ■     a  "I      ,T..^ 

or  ^x^  =  " 1  I  sni  X sin  2x  + sm  3x sm  4x  +  •  •  •    .  (B  ) 

4  ttLU         /  2  ^\VA      3V  4  J     ^     ^ 

The  differentiation  of  the  series  (A)  of  sines  will  give  a  series  in  wJiich  the  individual 
terms  do  not  approach  0  ;  the  differentiation  of  the  scries  (I?)  (if  cosines  gives 

I  TT  =  sin  X  +  \  sin  3 x  +  1  sin  .5 x  +  I  sin  7  x  +  •  •  • 

and  that  this  is  the  series  for  7r/4  may  be  verified  by  direct  calculation.   The  differ- 
ence of  the  two  series  (A)  and  (B)  is  a  Fourier  series 


/( 


IT       2V  cosox  "1      r  .  sin2x  T 

X)  =  -^  _  -^cosx  +  -^  +  .  .  .J  -  I  s,n  x  -  -^--  4-  . .  .J 


(C) 


which  defines  a  function  that  vanishes  when  0  <  x  <  tt  but  is  equal  to  —  x  when 

0  >  X  >    —  TT. 

174.   For  discussing  the  convei'gence  of  the  trigonometric  series  as  formallj' 
calculated,  the  sum  of  the  first  2  ?i  -|-  1  terms  may  be  written  as 

Sn=  ^    r"T-+  cos(<-x)  +  cos2(«-.r)  +  ■■■-{■  cos?i((  -  x)l/(/)c/i 

TT  i/O        l2  J 

sin  (2  n  -F  1) .t 

=  -    f       ; f(t)dt  =  -   i  ,   -/(■c  +  2») 

TT  «/  0  ,     .      t  —  X.  TT  J  -'- 


sin  (2  n  +1)m 


.    t-x 

I  sm 

2 


du, 


462  THEORY  OF  FUNCTIONS 

where  the  first  step  was  to  coinl>ine  o^.  cos  kx  and  hj^  sin  kx  after  replacing  x  in  the 
definite  integrals  (16)  by  t  to  avi)id  confusion,  then  summing  by  the  formula  of 
Ex.9,  p.  30,  and  finally  changing  the  variable  to  u  =  l{t  —  x).  The  sum  6'„  is 
therefore  represented  as  a  definite  integral  whose  limit  must  be  evaluated  as  n 
becomes  infinite. 

Let  the  restriction  be  imposed  upon /(.r)  tliat  it  shall  l)e  of  limited  variation  in 
tlie  interval  0<.r  <  2  7r.  As  the  function  f(x)  is  of  limited  variation,  it  may  be 
regarded  as  the  difference  P{x)  —  X{x)  of  two  positive  limited  functions  which 
are  constantly  increasing  and  which  will  be  continuous  wherever /(x)  is  continu- 
ous (§  1'27).  If /(.'■)  is  discontiiuious  at  x  —  Jq,  it  is  still  true  that/(x)  approaches 
a  Unfit,  which  will  be  denoted  Ijy  /(./•,-,  —  0)  when  x  approaches  x^  from  below  ;  for 
each  of  the  functions  P{x)  and  X{x)  is  increasing  and  limited  and  hence  each 
must  approach  a  limit,  and  f{x)  will  therefore  approach  the  difference  of  the  limits. 
In  like  manner /(/)  will  approach  a  limit /(.rg  +  0)  as  x  approaches  x^  from  above. 
Furthermore  as/(.r)  is  of  linfited  variation  the  integrals  required  for  .S'„,  «;-,  hi-  will 
all  exist  and  there  will  lie  no  difficulty  from  that  source.   It  will  now  be  shown  that 

lim.S,(x„)  =  lim  -  fl~^\f{x^,  +  2  u) '''"  ^~."  ^  ^^  "  du  =  -  [/(/^  +  0)-/(j,  -  0)]. 

n  =  X  n  =  00  TT  iJ  —    -  .sin  U  2 

This  will  show  that  tJie  ^erica  converges  to  tlie  fundion  icherever  the  funrtion  in  con- 
tinuous and  to  the  mid-point  of  the  break  ichererer  the  funrtion  is  discontinuous. 

T    .     ^/       ,    T    ,  sin  (2)1  +  1)"        .,       ,    -,    ,     "     .';in(2)i  +  l)u  sinA'u 

Let    f{x,,  +  2  u) : '—  =/(.fo  +  2  u) '—  =  F  (u) , 

sin  u  sm  u  u  u 

1     r"~^  „/  .  sinA-'.<  ,         1     r'',,  sintw  , 
then    S„  (.r„)  =  -        ,    -  F  («) du  =  ~         F  («) du.  -  tt  <  a  <  0  <  6  <  tt. 

TT  J-    "  U  TT  J  a  U 

As  fix)  is  of  limited  variation  provided  —  ir  <  a  ^  u  ^  ti  <  ir.  ho  must  /(/q  +  2  u) 
be  of  limited  variation  and  also  Fiu)  =  "/"/sin  u.  Then  F{u)  may  be  regarded  as 
the  difference  of  two  constantly  inci-easing  jiositive  functions.  r)r.  if  preferable,  of 
two  constantly  decreasing  jiositixe  functions  :  and  it  will  be  sutHcieiU  to  investigate 
the  integral  of  F{u)  u-^  slnku  under  the  hypothesis  tliat  F{u)  is  constantly  de- 
creasing.   Let  n  be  the  luimber  of  times  'l-Tr/k  is  contained  in  b. 

2  TT  -in-  2  HTT 

r'',       sin/i)/  ,          r^       pi-                   r'l-               r^'     ^, ,     sinA^u  , 
I     i-('M  '      -'da  =  /    '   +  /.,i  +  •  ■  •  +  /..,_,,.  +  f.^,,.^  (") ^^*^ 

=  \       +  +••+  /•-        -  du  +  /.„.  F{u) du. 

k 

As  F{u}  is  a  decreasing'  function,  so  is  u-  ^F{u/k).  and  hence  each  of  the  integrals 
whicli  extends  over  a  comi)lete  jieriod  2  tt  will  be  positive  liecause  tlie  neirative  ele- 
ments are  smaller  than  the  corresponding  positive  elements.  The  iiuci:ral  from 
2  nir/k  to  li  approaches  zero  as  /,•  becomes  infiinte.    Hence  for  large  values  of  k, 

r^\,,     Hxwku  ,  /'-/"^  ,,/(A  sin  w  ^.      i        ,  , 

(     /*  (") du  >  (         ^  [    ) ''"•        P  iixed  and  le.ss  than  n. 

Jo  u  J  ij  ^  k'     u 


SPECIAL   INFINITE  DEVELOPMENTS  463 

sin  ku 


Again,    I    F{u) du  =         +  + 

Jo  U  J  0  Jit  J"yTT 

r(->t-\)TT      /n\^\]^H  ph 


sin  ku  , 
du. 


Ht^Tf  all  llu"  tt'rni.s  except  tlie  lirst  and  last  are  iieii:ative  because  the  negative  ele- 
ments of  the  integrals  are  larger  than  the  positive  elciiieiits.    Hence  for  k  large, 

/^*_, ,  ^  sin  A-w  ,  /'(-^'-i)"„/w\ sin  «  ,  ^       , 

I    t  (u) a«  <   I  r  l~  ) du,        p  nxeu  and  less  than  n. 

t/o  u  Jo  \k/     u 

In  the  inequalities  thus  established  let  k  become  intinite.     Then  u/k  =  0  from 
above  and  F{u/k)  =  F{+  0).    It  therefore  follows  that 

„,   r*-''- '""sin  H  ,         ,.        r^'r,,  ,sinA7<  ,         „,      ^^^   /•-;"" sin  «  , 
i-(+0)  fZ«<l:m    I     i  (») (/m>F(+0)|        du. 

Althougli  p  is  lixed.  there  is  no  limit  to  the  size  of  the  number  at  which  it  is  fixed. 
Hence  the  inequality  may  be  transformed  into  an  equality 

/• ''  sin  A.'u  z' ^  sin  !<  tt 

liin    /    F(«)~— ^cZ«  =  F(+0)  f      -l^du  =  -  F{+0). 
k=A  J 0  u  Jo       u  2 

X^          sin  A'?*                           /* '-^  sin  H  tt 

F{u) du  =  F{-  0)  I du  =     F(-  0). 
....                 u                             Jo        u  2 

Hence  lim    f  ' F{u)^^^^ du  =- [F{+ 0)  +  F{- 0)] 

k-  =  -r.  J  a  U  2 

1     /> TT  —  -                         ciii  (2ji  4-  }^  u  1 

or  lim  ~    f  ,.„    -  fix,  +  2  u) \    ^    '    du  =  -  [f(x,  +  0)  +f{x,  -  0)]. 

n  =  /:   TT  J-  -  SHI  U  2 

Hence  for  every  point  .r^  in  the  interval  0  <  j'  <  2  tt  the  series  converges  to  the 
function  where  continuous,  and  to  the  mid-point  of  the  break  where  discontinuous. 
As  the  function /(,r)  has  the  period  2  tt.  it  is  natural  to  suppose  that  the  con- 
vei'Lience  at  .r  =  0  and  j  =  2  tt  will  not  differ  materially  from  that  at  ain^  other 
value,  namely,  that  it  will  be  to  the  value  I  [/(+  0)  A-f{2iv—  0)].  This  may  be 
shown  '\)\  a  transformation.    If  k  is  an  odd  integer,  2}(  -|-  1. 

sin  (2  n  -f-  1)  u  =  sin  (2  )i  +  I)  (tt  —  u)  =  sin  (2  n  -|-  1)  «', 

r        r^E-/  ,sin(2n-f  r))<                     f'V/   .,  sin  (2)i -|- 1)  m'      ,      t^ -n-,  ,        ,   ^n 
lim    I     F{u) r/(7  =  hm    I  F(u) du  = -F(ii  =  +  0). 

n  =  -x>  Jb  U  !(  =  ^  Jo  U'  2 

r"          sin  (2  )i  -I-  ]  1 '/  /^'^       r '"     tt 

Hence        lim    /     F{u)- — ! I^'-tZ«  =  lim    |     +    /     = -- [F(+ 0) -f-  F(7r  -  0)]. 


1    /^"'  sin  (2  ?i  -1-  1)  ?/ 

Now  for  j  =  0orj  =  27r  the  sum  ,^'„  =  -    I     /(^  w) ■ —  du,  and  the  limit 

TT  Jo  sin  u 

will  therefore  be  \  [/(+  0)  -|-/(2  tt  —  0)]  as  predicted  above. 

The  convergence  may  be  examined  more  closely.    In  fact 

,S„(x)=~       ^    -f{x  +  2u)- du  =  -   I        F{x.u) du. 

ttJ--  ii\r\u      u  7rJa(.T)  u 


464  THEOKY  OF  FUNCTIONS 

Suppose  0  <  a  ^  X  ^  /3  <  2  TT  so  that  the  least  possible  upper  limit  b  (x)  is  tt  —  i  ^ 
and  the  pjreatest  possible  lower  limit  a  (x)  is  —  J  a.  Let  n  be  the  number  of  times 
2  ir/k  is  contained  in  tt  —  ^  ^.    Then  for  all  values  of  x  in  a  ^  x  ^  )3, 


i<(x. -)  du-\-e<   \         F{x.u) di 

0  \     k/     u  Jo  u 


i: 


„  ,      u\  sin  u  , 

Fix,)  du  +  7],        p  <  71, 

k/      u 


where  e  and  rj  are  the  integrals  over  partial  periods  neglected  above  and  are  uni- 
formly small  for  all  x"s  of  nr  ^  x  s  ^  since  F{x,  w)  is  everywhere  finite.  This 
shows  that  the  number  p  may  be  cliosen  uniformly  for  all  x"s  in  the  interval  and 
yet  ultimately  may  be  allowed  to  become  infinite.  If  it  be  now  assumed  that  f{x)  is 
contiiuious  for  a  =  x  =  /3,  then  F(x,  ii)  will  be  continuous  and  hence  uniformly 
continuous  in  (x,  it)  for  tlie  region  defin^l  by  a  ^  x  ^  ^  and  —  ^  x  ^  w  s  tt  —  |  x. 
Hence  F(x,  u/k)  will  converge  uniformly  to  F{x,  +  0)  as  A;  becomes  infinite.    Hence 

F(x,  +  0)    I       du  +  e'  <   /         i'  (x,  M) du  <  F(x,  +  0)    (       du  -[-  y,' 

Jo        u  Jo  u  Jo        u 

where,  if  5  >  0  is  given.  K  may  be  taken  so  large  that  |e'|  <  5  and  |7j'|  <  5  for  A'  >  7v  ; 
with  a  similar  relation  for  the  integration  from  « (x)  to  0.  Hence  in  any  interval 
0<a^x^/iJ<27r  over  which  /(x)  is  continuous  .S„(x)  converges  uniformly 
toward  its  limit /(x).  Over  such  an  interval  the  series  may  be  integrated  term  by 
term.  If  /(x)  has  a  finite  number  of  discontinuities,  the  series  may  still  be  inte- 
grated term  l)y  term  throughout  the  interval  0  ^  x  ^  2  tt  because  -S'„  (x)  remains 
always  finite  and  limited  and  such  discontinuities  may  be  disregarded  in  integration. 

EXERCISES 

1.  Obtain  the  expansions  over  the  indicated  intervals.    Integrate  the  series. 
Also  discuss  the  differentiated  series.    Mal^e  graphs. 

/    ^       ■"'e-'^  11  1         ^  1         „  1  , 

(a) — cos  X  +  -  cos  2  X cos  3  x  -1 cos  4  x  —  •  •  • 

'  2siidi7r      2       2  5  10  17 

—  TT  t(.l    -f-   TT, 

1  2  ?  4 

+  -  sinx sin  2x4-  —sin  8x sin  4x  -1-  •  •  • . 

2  5  10  IT 

(/3)  I  TT,  as  sine  series,  0  to  tt.  (7)  -]  tt,  as  cosine  series,  0  to  tt, 

,  ^,     .  4  ri       cos2x       cos4x       cos  Ox  "i     „ 

(5)  smx  = ■  ■  ■    1  0  to  TT. 

TT  \_2         1  •  3  3-5  5-7  J 

(e)  cosx,  as  sine  series,  0  to  tt,  (f)  ("'.  as  cusinc  series,  0  to  tt, 

(tj)    X  sin  X,    —  TT  to  TT.  {0)    X  cosx,    —  TT  to  TT,  (l)    TT  -f  X,    —  TT  tO  TT, 

(k)  s'lnOx,  —  TT  to  TT.  (9  fractional,  (X)  cos(9x.  —  tt  to  tt.  (9  fractional, 

/     ^     r/    V  f  +  TT.    0  <  X  <  TT.        ,    ,     ^ ,    ,  f  \  IT.   0  <  X  <  I  IT,  .  .  ._ 

(/x) /(x)  = -^  '     •  ■     (v)  f(x)=i  '  ,        -  as  a  snie  series.  0  to  TT, 

^    ' -^  ^  '       1^0,  7r<x<27r,     ^  '  ■' ^  '       \  -  i  tt,  ,' 7r<x  <7r,  ' 

(o)   —  log  (2  sin'  I  =  cosx  4-    -  cos2x -f -^  cos3x  4- -  cos4x -f  •  •  • ,  0  to  tt, 


SPECIAL   INFINITE  DEVELOPMENTS  465 

(tt)  X,  —  ^tt  to  I TT,         (p)  sin  I  j,  —  i  tt  to  f  tt,         (o-)  cos  I  X,  —  I  tt  to  I  tt, 

(r)  from  (o)  find  expansions  for  lop;  cos  |x,  log  versa*,  log  tan  J  j.  Note  that  in 
these  cases,  as  in  (o),  the  function  does  not  remain  finite,  but  its  integral  does. 

2.  What  peculiarities  occur  in  the  trigonometric  development  from  —  tt  to  tt 
for  an  odd  function  for  which  f{x)  =f{Tr  —  x)?  for  an  even  function  for  which 
/(/)=/(7r-/)? 

3.  Show  that  f{x)=   >  o^. sm with  o/^.  =  -    |    f{x)sn\ dx  is  the  triso- 

-^  c  c  Jo  c 

nometric  sine  series  for /(x)  over  the  interval  0<x<c  and  that  the  function  thus 
defined  is  odd  and  of  period  2  c.  Write  the  corresponding  results  for  the  cosine 
series  and  for  the  general  Fourier  series. 

4.  Obtain  Xos.  808-812  of  Feirce's  Tables.    Graph  the  sum  of  Xos.  809  and  810. 

5.  Let  e  (x)  =  /(x)  —  IOq—  a^  cos  x  —  •  •  ■  —  a,i  cos  nx  —  l\  sin  x  —  ■  •  ■  —  b„  sin  nx 
be  the  error  made  by  taking  for/(x)  the  first  2  n  +  1  terms  of  a  trigonometric  series. 

1     /-  +  " 
The  mean  value  of  the  square  of  e{x)  is  —    I        [e(x)]-dx  and  is  a  function 

2  TT   J-TT 

F(«Q,  Oj,  •  •  • .  a„,  5j,  ■  •  ■ ,  b„)  of  the  coefficients.  Show  that  if  this  mean  square 
error  is  to  be  as  small  as  possible,  the  constants  a^,  a j,  •■-,«„,  5^,  •••,  6„  must  be 
precisely  those  given  by  (16) ;  that  is,  show  that  (16)  is  equivalent  to 

ca^      ca^  da,,      cb^  cbn 

6.  By  using  the  variable  X  in  place  of  x  in  (16)  deduce  the  equations 

/(x)  =  —   f /(X)co^0(\-x)(?\  + -V    r'7(X)cosA-(X-x)d\ 
=  J- V    r^f{\)(j^^'^-''''d\  =  -^  2  t^^-"'  r""/(J-)e=^*'-'-''(7x  ; 
and  hence  infer         /"(x)  =  "V  (^c- '■'■',         a^- =  —    |      /(x)e±^-"'dx. 

^  2  TT  v'o 

7.  Without  attempting  rigorous  analysis  show  formally  that 


/: 


4>{a)da  =    lim  [ \-  <p{—  ?i  •  A(i)A(^r  +  0(—  7i  +  1  •  Aa)Acr  H \-  4>{—\  •  Aa)Acr 

+  (/>(0-  A<i-)A(r  +  </!>(l  ■  Aa)Acr  +  •  •  •  +  ^(n-  Anr)A(r  +  •  •  •] 


a\  a 
c 


=    lim   V  0(A-- A(i-)Aa- =  lim  V0(/^~)- 

\a  =  0  ■^  r  =  X  -^       \     C  / 

Show  /(x)  =  :/7.2 /_/(x)c"''^''''"'"''^x  =  J^;g X/w*^"^ 

is  the  expansion  of /(x)  by  Fourier  series  from  —  c  to  c.    Hence  mier  thai. 

/(x)  =  —    f     r'/(X)e^«(A--')'(ZXrfa=  lim  --- V    f  \/-(X)c^  >  ^""'"dX '' 


466  THEOKY  OV  FUNCTIONS 

is  an  expression  for/(x)  as  a  donble  integral,  which  may  be  expected  to  hold  for 
all  values  of  x.    Reduce  this  to  the  form  of  a  Fourier  Integral  (Ex.  15,  p.  377) 

/(x)  =  i   f  r'    f{\)coiia{\-x)d\d(x. 

TT  Jo       J  -  r. 

8.  Assume  the  possibility  of  expanding /(x)  between  —  1  and  +  1  as  a  series  of 
Legeudre  polynomials  (Exs.  1:3-20,  p.  252,  Ex.  10,  p. 440)  in  the  form 

fix)  =  a^V,^{x)  +  aj"^  (x)  +  ((.P.  (x)  +  •  •  •  +  «„P„(x)  +  •  •  •  . 

By  the  aid  of  Ex.  19,  p.  253,  determine  the  coefficients  as  a^  = i    /(x)  Pyt(x)  dx. 

2       J-i 
For  this  expansion,  form  t3(x)  as  in  Ex.5  and  show  that  the  determination  of  the 
coefficients  cti  so  as  to  give  a  least  mean  square  error  agrees  ^^■i^!l  the  determi- 
nation here  found. 

9.  Note  that  the  expansion  of  Ex.  8  represents  a  function  /(x)  between  the 
limits  ±  1  as  a  polynomial  of  the  nth  degree  in  x,  plus  a  remainder.  It  may  be 
shown  that  precisely  this  polynomial  of  degree  n  gives  a  smaller  mean  scjuare  error 
over  the  interval  than  any  other  polynomial  of  degree  n.    For  suppose 

Un  (j)  =  Co  +  c^x  +  •  •  ■  +  c„x"  =  h^  +  bj\  +  ■■■  +  b„}\ 

be  any  polynomial  of  degree  n  and  its  equivalent  expansion  in  terms  of  Legeudre 
polynomials.  Now  if  thec's  are  .so  determined  that  the  mean  value  of  [/(■'')  —  f/n(J")]^ 
is  a  mininuun,  so  are  the  b's,  which  are  linear  homogeneous  functions  of  the  c's. 
Hence  the  //s  nuist  be  identical  with  the  *rs  above.  Note  that  whereas  t'.ie  Maclaurin 
expansion  replaces /(x)  by  a  pcjlynomial  in  x  whic'.i  is  a  very  good  approximation 
near  x  =  0,  the  Legeudre  expansion  replaces  /{x)  by  a  polynomial  which  is  the 
best  expansion  when  the  whole  interval  from  —  1  to  +  1  is  considered. 

10.  Compute  (cf.  Ex.  17,  p.  252)  the  polynomials  7\  =  x,  P.  =  -  ;i  +  |  x^, 

P     _  -1   r  4-   5  r3  P     —  »  15    ,■2    i     3  ">   /.i  p     _  ijs  .»  3  5  j.3    i     6  3  ^.5 

'3   —  2  '      2         '  4   —    K  1  '        H  ^  ."»   —      8  4    "^'      T^      g    "^    • 

r^  ■  2  /       (')  \      2 

Compute    I     X'  sinTrxfZx  =  0,  —  I  1 I.  0.  -  .0  when  I  =  4,  3,  2,  1.0.    Hence  show 

that  the  polynonual  of  the  f(jurlli  degree  which  best  reiiresents  sin  ttx  from  — 1 
to  +  1  reduces  to  degree  three,  and  is 

sin  TTX  =  -^  X  -  -  C-  =  l]  i-  X-'  -  ^  A  =  2.0i)x  -  2.80x3. 


TT  TT  \7r- 

Show  that  the  mean  .sijuare  error  is  0.004  and  compare  with  that  due  to  Maclaurin's 
expansion  if  tlie  term  in  x'*  is  retained  or  if  tlie  term  in  x'^  is  retained. 

11.  Expand  .sin  ^  ttx  =  ^']  I\  -  ^''^  P-^-  -  \]  l\  =  1.55:!x  -  0.5()2.r3. 

2  TT-  TT-    \7r-  / 

12.  Expand  from  —  1-  to  +1,  as  far  as  indit'aled,  these  functions  : 

((f)   cos  TTX  to  P^,  (/3)   f-  t<>  /'..  (7)log(l  +  .r)      toP^, 

(5)    \^l  -  X-      to  P^,  (e)   cos-lx         to  P,.  (f)    tan-'x  toP-, 


(^)  -^_       tor,,  (^)        J=.     toP„  (0^^=         toPg. 

\  1  +  X  ^  1  —  ./■-  Vl  +  x^ 

AVhat  simplilications  occur  if /'(x)  is  odd  or  if  it  is  even  ? 


SPECIAL   INFINITE  DEVELOPMENTS  467 

175.  The  Theta  functions.  It  has  been  seen  that  a  function  with  the 
period  2  ir  niay  be  expanded  into  a  trigonometric  series  ;  that  if  the 
function  is  ochl,  tlie  series  contains  only  sines ;  and  if,  furthermore, 
tlie  function  is  symmetric  with  res])ect  to  x  —  \7r,  the  odd  multiples 
of  the  angle  will  alone  occur.    In  this  case  let 

/(.r)  =  2  \_<i^^  sin  X  —  a^  sin  3  a'  H +  (-  1)"  a„  sin  (2n  +  l)x  ^ ]. 

As  2  sin  nx  =  —  i  (e'"' —  e~""),  the  series  may  be  written 

fix)  =2^  (-  l)"a.„  sin  (2  r.  +  l).^"  =  -  ^  X  (-  l)^^e(^"  + 1)"',  «_„  =  ./„_,. 

0  -   -A 

This  exponential  form  is  very  convenient  for  many  purposes.  Let  Ip 
be  added  to  x.    The  general  term  of  the  series  is  then 

Hence  if  the  coefficients  of  the  series  satisfy  a^^_-^e~""P  —  c/„,  the  new 
general  term  is  identical  Avith  the  succeeding  term  in  the  given  series 
multiplied  b}^  —  ife'-'-".    Hence 

/(.«  +  Ip)  =  -  e^e-^-  /(./•)     if     a,_,  =  ay^-i'. 

The  recurrent  relation  between  the  coefficients  will  determine  them 
in  terms  of  a^.    Eor  let  y  =  e~f.    Then 

The  new  relation  on  the  coefficients  is  thus  (;ompatiV)le  with  the  original 
relation  ((_j^  =  «„_i.    If  a^  =  q*,  the  series  thus  becomes 

/(.r)  =  2y'sina--2-'/sin3a;H \-(~\)"2qi^-"^'^''shi(2n  +  l)x  +  ■  ■  ■, 

f(x  +  2  TT)  =f(x),     f{x  +  tt)  =  -/(,r),     /(,.  +  Ip)  =  -  q-h^-'^'f(.r). 

The  function  thus  dehned  formally  has  im})ortant  properties. 

In  the  first  place  it  is  important  to  discuss  the  convergence  of  the 
series.    Apply  the  test  ratio  to  the  exponential  form. 


"«  +  i/"«  =  '/V- 


v^_„  =  (\ 


,1  il  .j~  'Ixi 


Eor  any  x  this  ratio  will  approach  the  limit  0  if  y  is  numerically  less 
than  1.  Hence  the  series  converges  for  all  values  of  x  provided  \q\<.  1- 
Moreover  if  \x\  <  }^(1,  the  absolute  value  of  the  ratio  is  less  than  \qf"e''', 
Avhich  approaches  0  as  n  becomes  infinite.  The  terms  of  the  series 
tlierefore  ultimately  become  less  than  those  of  any  assigned  geometric 


468  THEORY  OF  FUXCTIOXS 

series.  This  establishes  the  uniform  convergence  and  consequently  the 
continuity  of.  f{x)  for  all  real  or  complex  values  of  ./■.  As  the  series  for 
/' (x)  may  be  treated  similarly,  the  function  lias  a  continuous  derivati\e 
and  is  everywhere  analytic. 

By  a  change  of  variable  and  notation  let 

^00=/(^;V       y  =  '^'"'^  (19) 


r.     1      ■        TT'/  r,     '^      ■      3  ITU  ,      2^=.      .      5  TTU  ,^^ 

//(»)  =  2y*  sni--—- 2^5  Sin ---r  + -*'/ *   «i"  77— •      (20) 

The  function  Il(jt),  called  eta  of  u,  has  therefoi-e  the  pro])erties 

II(h  +  2  K)  =  -  H(u),  H(u  +  2  iK')  =  -  7-ir^"'//(^/),    (21) 

inrr 

■  H(u  +  2  mK  +  2  InK')  =  (—  l)"'  +  "'y-"e~"^^" "//(?/),      7»,  ?i  integers. 

The  quantities  2  A'  and  2  //v '  are  called  the  periods  of  the  function.  They 
are  not  true  periods  in  tlie  sense  that  2  tt  is  a  period  off(x) :  for  Avhen 
2  K  is  added  to  v,  the  function  does  not  return  to  its  original  value,  but 
is  changed  in  sign ;  and  Avhen  2  IK'  is  added  to  u,  the  function  takes 
the  multiplier  written  above. 

Three  new  functions  will  be  formed  by  adding  to  u  the  quantity  A' 
or  iK'  or  K  -\-  IK',  that  is,  the  half  periods,  and  niaking  slight  changes 
suggested  by  the  results.  First  let  H.^{i()  =  H(i(  -j-  K).  By  substitution 
in  the  series  (20), 

H^ {u)  =  2  y ^"  cos  —  +  2  7?  cos  -y-^  +  2  V  ^  cos  — .-  +  •  •  ■ .     ( 22 ) 

By  using  the  properties  of  //,  corresponding  properties  of  7/^. 

H^0(  +  2  A)  =  -  //j  U).  11^  („  +  2  iK')  =  +  'r''~  "  "JI,  (<'),  (-'^) 

are  found.    Second  let  iK'  be  added  to  (/  in  11(1'}.    Then 

is  the  general  term  in  the  exponential  development  of  Ihu  +  iK') 
apart  from  the'  coctticicnt   ±  /.    Hence 

H{,i  +  iK')  =  /  y(-  i)Y'""^r^A-",;-"-'A-" 


h~''-'^'"t. 


(-1)7 


SPECIAL  INFINITE  DEVELOPMENTS  4G9 

1      *^  CO  q  ni 

Let  ©(?/)  =  -  l'/e^'"H{u  +  iA'')  =  ^  (-  Ij^/'e'""^'"  • 

The  development  of  ©(/<)  and  further  properties  are  evidently 

0  (u)  =  1  -  2  y  cos  -^^  +  2  y*  cos  y^  -  2  ./  cos  y^  +  •  •  • ,      (24) 

®(u  +  2K)  =  &(h),  ®(u-\-2  iK')  =  -q-h~'^"®(H).         (25) 

Finally  instead  of  adding  K  +  IK'  to  ?^  in  H  (ii),  add  A'  in  0  (^/). 

0,(^0  =  1  +  2  ./  cos  YY  +  2  7^  cos  — -  +  2  ./  cos  y^  +  •  •  • ,      (26) 

0^(^.  +  2  A)  =  0/^/),         0/«  +  2  .-A')  =  +  y- VX^«  0^(,<).         (27) 

For  a  tabulation  of  properties  of  the  four  functions  see  Ex.  1  below. 

176.  As  H  {u)  vanishes  for  u  =  0  and  is  reproduced  except  for  a 
finite  multiplier  when  2  mK  +  2  nlK'  is  added  to  w,  the  table 

H (xi)  =  0  for  u  =  2  ;« A  +  2  nlK', 

H^(u)  =  0  for  u  =  (2  m.  +  1)  AT  +  2  nlK', 

0  (?<)  =  0  for  «  =  2  /y^  A  +  (2  M  +  1)  tA', 

®^(u)  ^  0  for  w  =  (2  m  +  1)  A  +  (2  7i  +  1)  iK', 

contains  the  known  vanishing  points  of  the  four  functions.    Now  it  is 
possible  to  form  infinite  products  which  vanish  for  these  values.    From 
such  products  it  may  be  seen  that  the  functions  have  no  other  vanish- 
ing points.    ^Moreover  the  products  themselves  are  useful. 
It  will  be  most  convenient  to  use  the  function  ®^{<()-    Now 

eK^  '  =  —  y<-«+l)j  —    X    <    ?i   <   CC  . 

Hence         e^"  +  ry-(-"+i)     and     g-^"  +  .y-(2»+i),         n  ^  0, 

are  two  expressions  of  which  the  second  vanishes  for  all  the  roots  of 
0|(//)  for  which  n  ^  0,  and  the  first  for  all  roots  with  n  <  0.    Hence 

TT  =  C  fl'  (l  +  rf+^e'^")  (l  +  r/"+ie-^) 

is  an  infinite  product  which  vanishes  for  all  the  roots  of  0j(«)-  The 
product  is  readily  seen  to  converge  absolutely  and  uniformly.  In  par- 
ticular it  does  not  diverge  to  0  and  consequently  has  no  other  roots 
than  those  of  ®^(i()  above  given.  It  remains  to  show  that  the  product 
is  identical  with  @^(t()  with  a  proper  determination  of  C. 


470  THEOEY  OF  FUNCTIONS 

in 

Let  6j(ii)  be  written  in  exponential  form  as  follows,  with  z  =  e^    : 

0(2)  =  e,{u)  =  1  +  q  (^z  +  -^  +  q*  (^Z^  +  -^  +  .  .  .  +  q--  /^^  +  i\  +  .  .  . , 
^P{Z)  =  C'-1TT(")  =  (1  +  (1Z){1  +  q^z){l  +  q^z)-  •  ■  (1  +  q"->^-^z)-  ■  ■ 


A  direct  substitution  will  show  that  0  {q-z)  =  ry- 1^:- 1^  (z)  and  \(/  {q-z)  =  q-'^z-'^^  {z). 
In  fact  this  substitution  is  equivalent  to  replacing  u  by  u  +  2  jA''  in  Gi.  Next  con- 
sider the  first  2  ?i  terms  of  \p  [z)  written  above,  and  let  this  finite  product  be  i/'„(2). 
Then  by  substitution 

((/«  +  qZ)-^„{qh)  =  (1  +  r/«  +  l2)^„(r). 

Now  i/'n  (2)  is  reciprocal  in  2  in  such  a  way  that,  if  multiplied  out, 

V'n  (2)  =  «o  +  «i(2;  +  ;)  +  "•..  (2'  +  ;^)  +  •  •  •  +  ««(2"  +  ^\         ««  =  qn\ 

n  n 

Then         (ry2 n  +  ,^2)  ^  (, .  (,/2,v--  +  ,/- :;  /.- ')  =  (1  +  q"- "  + 12)  2  r(,  (2'  +  2- '), 

0  0 

and  the  expansion  and  equation  of  coefficients  of  z'  aives  the  relation 

"<■=<'»•- 1^ _      .,      ,^ "!•       "l  =  "ij~JZi 

/:  =  U 

From    (1,1  =  (j"',         "0  —        "  """  —      >         "t 


/(•=  1 


TT  (1-7-^)  TT  (1-'/-^) 

A  =  1  1=1 

Now  if  n  be  allowed  to  become  infinite,  each  coefficient  (ii  approaches  the  limit 

lim  Ui  =  ^  .         C  =  fr  (1  -  7- «)  =  (1  -  7-)  (1  -  7')  (1  -  ^y')  •  •  •  • 
0  1 

Hence  GJm)  =  TT  d  -  7-")  ■  TT  U  +  7""  + ^e^  U  +  7-" +  't'  '■■     A 

1  0 

provided  the  limit  of  i/',, (2)  may  be  fdund  by  taking  the  series  of  the  limits  of  the 
terms.    The  justification  of  tins  proi'css  would  be  similar  to  that  of  §  171. 

TIh'  products  for  0.  H^,  If  may  Lo  obtained  from  tliat  for  0^  liy  siili- 
tractiiig  K,  IK',  K  -f-  iK'  from  n  and  making  the  needful  sliglit  altera- 
tions to  conform  with  the  deiinition.s.  The  products  may  l)e  converted 
into  trigonometric  form  Ity  multi])lying.    Then 

n{u)  =  c 2  v'-  sm  J-f.  fr  (1  -  2  r"  ^'<«  ?7^''  +  '■'")■        i^^) 


■1  K   1    \  '  2  K 


SPECIAL   INFINITE  DEVELOPMENTS  471 

H^i,)  =  C  2  >/^  cos  ^^  TT  ("l  +  2  r'^  cos  |-^  +  >A,  (29) 

0(^0  =  C  fr  (l  -  2  ./"  +  ^  cos  1^  +  ./«  +  ^),  (30) 

©X^/)  =  C  fr  ("l  +  2  ./^ « +1  cos  1^'  +  '/ "  +  ^ V  (31) 

C  =  TT  (1  -  y  ^ ")  =  (1  -  >f)  (1  -  ,f)  (1  _  ./)  . . . ,  (32) 

//^O)  =  r  2  y ^  fr  (1  +  v^ " ;)^        0  (0)  =  r  TT  ( 1  -  <f "  +  'y, 

1  0 

//'(O)  =  C  2  .y^  ^.  IT  (1  -  r"f,         ®,(0)  =rTl  (1  +  y-^"  +  iy^. 

Tilt'  value  of  If'(O)  is  found  by  dividing  //('')  by  ?/  and  letting  u  =  0. 
Then 

II '(0)  =  ^^,  7/^0; 0(0) 0^(0;  (33) 

follows  by  direct  substitution  and  cancellation  or  condjination. 
177.   Other  functions  may  Ije  built  from  the  theta  functions.    Let 

'       &{K)       0^0)'  '        0,(0/  \A-       //,(0)'    ^"  ^ 

1    //(//)                            IP //,(//)             ^               /7^0i''")       ,.>- 
sn  (<  =  — ^ — 7— J  ''i^  "  =  A    ; '  dn  ^/ =  VA-  — ^ (3.)) 

The  functions  sn  i/,  en  ;/,  d]i  u  are  called  elliptic  functions*  of  i>.  As  7/ 
is  the  only  odd  theta  function,  sn  i'  is  odd  but  en  u  and  dn  u  are  even. 
^1//  fl/ri-r  fii/trftn/)s  lid  re  tiro  octiiiil  jh'i'IoiIs  in  the  same  sense  that  sin./' 
and  cos  ./■  have  the  }»eriod  2  tt.  Thus  dn  //  has  the  })criods  2  7v  and  4  IK' 
liy  (25),  (27):  and  sn  u  has  the  jieriods  4  K  and  2  /7v'  by  (~-i),  (21). 
That  en  u  has  4  7v  and  2  7v  +  2  IK'  as  jieriods  is  also  easily  verified. 
The  values  (if  f  Avhicli  make  the  functions  vanish  are  known:  they  are 
those  which  make  the  numerators  vanish.  In  like  manner  the  values 
of  u  for  which  the  three  functions  Ijecome  infinite  are  the  known  roots 

of  ©((/). 

If  q  is  known,  the  values  of  vT  and    vZ''  may  be  found  from  their 
definitions.    (.'(Uiversely  the  ex})ression  for   vA-', 

^^'   -  ^W)  -l  +  2y  +  2,^  +  2,«+...'  ^^^^ 

*  The  study  of  the  elliptic  fuuctious  is  continued  in  Chapter  XIX. 


472  THEORY  OF  FUNCTIONS 

is  readily  solved  for  q  by  reversion.    If  powers  of  q  higher  than  the 
first  are  neglected,  the  approximate  value  of  q  is  found  by  solution,  as 

1  1  _  VA^  _   q  +  /  +  - 


2l+V/.-'      1 --''/  + 


q-2q'-{-bq^  + 


1 1  -  A/.'     2  /I  -  Va^Y    15  /I  -  V;^'V 

Hence      q  —  7. ^  +  7?,    ?=  )  +  "^    1=  I  +  •  •  ■  ("J  0 

is  the  series  for  q.  For  values  of  A-'  near  1  this  series  converges  with 
great  rapidity;  in  fact  if  A-''^  ^  \,  k'  >  0.7,  "V^  >  0.82,  the  second  term 
of  the  expansion  amounts  to  less  than  1/10^  and  may  l)e  disregarded 
in  work  involving  four  or  iive  figures.  The  first  two  terms  here  given 
are  suflBcient  for  eleven  figures. 

Let  />  denote  any  one  of  the  four  theta  series  //,  H^,  0,  0^.   Then 

^^(U)  =  <{>(,)  =  ^  />„,'',  Z=:C-K"  (38) 

may  be  taken  as  the  form  of  develo])nient  of  >'/-;  this  is  merely  tlie 
Fourier  series  for  a  function  with  period  2  A'.  But  all  the  theta  func- 
tions take  the  same  multiplier,  except  for  sign,  when  2  iK'  is  added  to  u; 
hence  the  squares  of  the  functions  take  the  same  multiplier,  and  in  par- 
ticular 4>(q'^)  —  q~-z~-<f>(z).    Apply  this  relation. 

It  then  is  seen  that  a  recuri-cut  I'clation  between  the  coefficients  is  found 
which  will  deternnne  all  the  evi'U  coefticients  in  terms  of  //^^  and  all  the 
odd  in  terms  of  b^.    Hence 

'^-('0  =  h^ (■-)  +  ^^0),         f>,,  ^.  constants,  (38') 

is  the  expansion  of  any  ''/'-  or  of  any  function  which  may  be  develo})ed 
as  (38)  and  satisfies  <ji(q'z)  =  q~-r:~-(f)(x).  ^Moreover  $  and  ^  are  iden- 
tical for  all  such  functions,  and  the  only  difference  is  in  the  values  of 
the  constants  A^  fi^. 

As  any  three  tlieta  functions  satisfy  (38')  with  different  values  of  the 
constants,  the  functions  $  and  ^  may  l)e  eliminated  and 

n''}{(n)  +  /3''K:(")  +  r'^iOn  =  o, 

where  a,  /?,  y  are  constants.  In  words,  the  squares  of  any  three  theta 
functions  satisfy  a  linear  hdinugeneous  ecjuation  with  constant  coeffi- 
cients. The  constants  may  be  determined  l)y  assigning  ]»articular  values 
to  the  argument  //.    For  example,  take  //.  11^.  0.    Then* 

♦For  brevity  the  pareuthesis  abuut  the  argiuiieuts  uf  a  fuuetion  will  frequently  be 
omitted. 


SPECIAL   INFINITE  DEVELOPMENTS  473 

aH\it)  +  ftHf  (u)  =  y®-(if),  y8i7f0  =  y©-0,  nll-R  =.  y©-A', 

0-A'  H-(>')        0-'O  H{(u)       ,  ,  ,  , 

/rtC-  0Vi  +  7/fo  -0%  =  1'  "'■ '"  "  +  ■■■'  «  =  1-        (39) 

By  treating  //,  0^,  0  in  a  similar  manner  may  be  proved 

//-  sn-  If  +  dn-u  =  1     and     Jr  +  /.■'-  =  1.  (40) 

The  function  >'j(i/)i'j(i/  —  a),  -where  a  is  a  constant,  satisfies  the  rela- 
tion <j>(<fz)  =  rj~-z~"C<j>(z)  if  log  C  =  irra/K.  Eeasoning  like  that  used 
for  treating  {^-  then  shows  that  between  any  three  such  expressions 
there  is  a  linear  relation.    Hence 

aH(ii)If(u  -  a)  +  /3nju)II^(H  -  »)  =  y&(,/)&(t(  -  a), 

n  =  0,  /?//j (0)  //j (^f)  =  y0 (0; 0 {n), 

V  =  K,  all^  (0)  H^  (a)  =  y©^  (0)  ©,  (a), 

&0®,0(d,<iH(ii)H(u  -  (i)        ©-0  H,(u)II^(ii  —  (i)  _   ©0^  Il^a 
ir^O@a@(it)&(u  -  a)  llfO    0 (//)©(;/  -  a)     ~  II ^  ^  ' 

or  dn  (I  sn  u  sn  (u  —  c)  +  en  ti  en  (a  —  «■/)  =  en  <(.  (41  j 

In  this  relation  re])lace  o  Ijy  —  r.    Then  there  ivsults 

en  i/  cii(i(  +  r)  +  sn  u  dn  r  sn  (/i  -\-  r)  =  en  r, 

or  en  v  en  (u  +  r)  -f  sn  v  dn  u  sn  (k  +  '")  =  en  «, 

en- 1/  —  en-  (•  =  sn'-  v  —  sn'-  tr  ,  ,^^ 

and  sn(^;  +  r)  = —  ,  (42) 

sn  r  en  »  dn  <<  —  sn  «  en  r  dn  (• 

by  symmetry  and  by  solution.  The  fraction  niay  lie  reduced  by  multiply- 
ing numerator  and  denominator  by  the  denominator  -with  the  middle 
sign  changed,  and  by  noting  that 

sn'-  c  en-  u  dir  ti  —  sn'-  tt  eir  c  dn'-  r  =  (sn-  v  —  sn'-  (/ )  ( 1  —  /.•'-  sn-  ii  sn'-  r). 

siw/ en  r  dn  r -f- sn  r  en /?  diw^  ,,„, 

Then  sn  (>>  +  r)  = ~~ 3 '  (43) 

^  1  —  Jr  H\r  u  sn-  r  ^     ^ 

,  ,  sn  I)  en  r  dn  r  —  sn  c  en  ^/  dn  n 

and  sn(»  —  r)  = ~ — — , ^ ) 

1  —  /.-sn-  ^^  sn-  /• 

1  ^  ^  ^  2sn  r  en  « dn  k  .... 

and  sn(*/  +  r)  —  sn((/  —  r)  = -; — :, ^^  •  (44) 

1  —  A-sn-  «sn-  /•  ^ 

The  last  result  may  be  used  to  differentiate  sn  n.    For 

sn(//  +  \n)  —  sn  K  _  sn  i-  A^^   cnf/^  +  I  \ii)({\\(ii  +  i^  A/') 
A/^  ~     \  \ii     1  —  /.•'-  sn- 1  \u  sn'-(//  +  \\ii)' 

(J                            ,                         , .      sn  //  , ,  _ 

-—  sn  ;/  =  a  on  u  dn  ?/,  a  =  lini •  (io) 


4T4  THEORY  OF  FUNCTIONS 

Here  g  is  called  the  mult'qji'wr.    By  definition  of  sn  u  and  by  (33) 

^"i/,(0)  0(0)  ~2A'®^^^>  ^^-"^ 

The  periods  2,  K,  2iK'  have  been  independent  up  to  this  point.  It  will, 
however,  be  a  convenience  to  have  g  —  1  and  thus  simplify  the  formula 
for  differentiating  sn  //.    Hence  let 

g  =  l,         ^^  =  ©,(0)  =  l  +  2y  +  2,/  +  ---.  (46) 

Now  of  the  five  quantities  K,  K',  k,  k',  q  only  one  is  independent. 
If  q  is  known,  then  A:'  and  K  may  be  computed  by  (36),  (46) ;  /.■  is  de- 
termined by  1^'  +  /.:"  =  1,  and  K'  by  ttK'/K  =  -  log  7  of  (19).  If,  on  the 
other  hand,  /.;'  is  given,  q  may  be  computed  l)y  (37)  and  then  tlie  other 
quantities  may  be  deternuned  as  before. 

EXERCISES 

in                                    iiT 
_  1 «  u 

1.  With  the  notations  X  =  f/    *e    -^  ,/j.  =  q-'^e    ^     establish: 

//(-  u)=-  II{u),       U{u  +  -A  K)=-  JI{u),  II{u  +  2iK')=-  ij.TI{u), 

TI^  (-  u)  =  +  i/j  (»),     H^ {u  +  -2  K)  =  -  7fj  («),  ir^  {u  +  2  IK')  =  +  f,l[^  (h), 

e(-M)  =  +e(M),       Q{u  +  2K)  =  +  e{u),  e{u  +  2iK')  =  -  ij.e{u), 

ei(-M)  =  +ei(M),  Q^{u+2K)  =  +e^{ii),  Q^{u +  2  1K')=+  ^Q^{u), 

II {u  +  K)  =  +  11^  («),  //  (»  +  IK')  =  /xe  (m),  II {u  +  K  +  iK')  =  +  XGi  (h), 

II ^{u  +  A')  =-  II (u),  II ^{u  +  IK')  =  +  Xei(u),  II^{u+  K  +  iK')  =—  iXG(w), 

e  (u  +  K)  =  +  Oj  (it).        0  {u  +  iK')  =  i\//  (m),  e  (({  +  A'  +  iK')  =  +  X/fj  («), 

Bj  ((i  +  A)  =  +  e  (w),  Bj  ((i  +  iK')  =  +  \II^  {u),  Bj  (u  +  K  +  iK')  =  +  i\II{u). 

2.  Show  that  if  u  is  real  and  q  ^  jl,  tlie  tirst  two  trigononu'tric  terms  in  tlie 
series  for  //,  7/^,  B,  Bj,  give  four-i)laee  aecuraey.  Sliow  that  with  q  ^  0.1  tliese 
terms  give  about  six-place  accuracy. 

3.  I'se _  --'  - =  (I  sin  n-  +  q-  sin  2(1-  +  q^  sin  8  tr  -f-  •  •  •  to  prove 

1  —  2  r/  cos  a'  -f  (/- 

t.       TTK              ,     .       2  -RH             ,,      .       3  TTU  \ 

f/sm      -       r/- sm  -       -       (/•*  sm \ 

A'  A  K 

4.  I'rovt!  tlie  double  periodicity  of  en  u  and  sliow  that  : 

•■I'  "  ,  .  ,.,s  1  ,  ,.       .  ,^,,         dn  u 

sn  (»  4-  A  )  =  —    ,         sn  (11  +  th  )  = ,         sn  (h  +  J\  +  th  )  = , 

dn  u  k  sn  a  k  en  /( 

—  A'' sn  »  ,         ,,,,^       —ii\nu  ,  ,,       .,.,,        —  ik' 

cn(« -f-  A)  =  -      -   '     ,         cn(M. -f- t/v  )  = ,         cn(u  +  Jv  -f  <A  )  = , 

dn  H  k  sn  «  A' en  » 

1    /  r-  A;'  ,     ,         ._,,„  .cnii  ,     ,         ,,       .  „,,       .,,sn(f 

an  (h  -f  a  )  =  -       ,         dn  (h  4-  <7v  )—  —  i ,         dn  (m  -\-  K  ■\-  %K)  —  ik 

dn  M  sn  M  en  u 


SPECIAL   INFINITE  DEVELOPMENTS  475 

5.  Tabulate  the  values  of  sn  u,  en  u,  tin  u  at  0,  A',  iK',  K  +  IK'. 

6.  Compute  k'  and  k-  for  q  =  \  and  q  =  0.1 .   Hence  show  that  two  trigononietric 
terms  in  the  theta  series  give  four-place  accuracy  if  k'  ^  l. 

-    ,,  ,  ,       en  u  en  r  —  sn  u  sn  w  dn  li  dn  v 

7.  Prove  en  («  +  r)  =  — 


and  dn  (w  +  v)  = 


1  —  A;2  sn^  u  sn^  v 

dn  zt  dn  v  —  k!^  sn  w  sn  v  en  it  en  u 
1  —  k^  sn-  u  sn^  v 


8.  Prove  —  en  «  =  —  snudnzt,         — {\\\u  =— k'^^nucwii         (/ =  1. 
(Ill  du 


9.  Prove  sn-iw  :=  f   — -^  from  (45)  with  g  =  1. 


du 
V(l-u2)(l-A;2u2) 

10.  If  r/  =  1,  compute  k,  k',  K,  K\  for  q  =  0.1  and  q  =  0.01. 

11.  If  (/  =  1,  compute  k',  q,  K,  K\  for  A:'-  =  |,  f,  ^. 

12.  In  Exs.  10,  11  write  the  trigonometric  expressions  which  give  sn  u,  en  u,  dn  u 
with  four-place  accuracy. 

13.  Find  sn  2  ii,  en  2  «,  dn  2  ?/,  and  hence  sn  },  u,  en  I  u,  dn  ?,  w,  and  show 

snJir  =  (l+ A-')^2,         cn-|Jv=  VA'(l+/c')-i,         dn  |  A' =  VP. 


14.  Prove  —  A  I  sn  m  dn  =  log(du  u  +  ken  u)  ;  also 


e-^(0)7/(M  +  ,,^ //(((_«)  =  e2(r()//2(M)-  //2(rt)e2(«), 

02  (0)  e  (W  +  «)  e  {«  -  a)  =  92  («)  02  {<()  -  7/2  (i<)  //2  ((,)  . 


CHAPTER  XVIII 

FUNCTIONS  OF  A  COMPLEX  VARIABLE 

178.  General  theorems.  The  (•omplex  function  it  (x,  y)  +  iv  (.r,  ?/), 
where  ?/  {x,  //)  and  r  (./■,  _//)  are  single  valued  real  functions  continuous 
and  dift'erentiable  })artially  with  respect  to  a-  and  //,  has  been  defined 
as  a  funcjtion  of  the  complex  variable  z  =  ,r  +  HI  '^^'^'sn  and  only  when 
the  relations  ?'^  =  r'^  and  ^i',,  =  —  '"',  are  satisfied  (§73).  In  this  case 
the  function  has  a  derivative  with  res})ect  to  z  which  is  independent 
of  the  way  in  which  A,v  ap})roaches  the  limit  zero.  Let  iv  —  f{z)  be  a 
function  of  a  complex  variable.  Owing  to  the  existence  of  the  deriva- 
tive the  function  is  necessarily  continuous,  that  is,  if  e  is  an  arbitrarily 
small  positive  number,  a  number  8  may  be  found  so  small  that 

|/C^)-/C^o)l<^     ^^'I'en     \z-z^\<^,  (1) 

and  nioi'cover  this  relation  holds  uniformly  for  all  points  z^  of  the 
region  over  which  the  function  is  defined,  provided  the  region  includes 
its  bounding  curve  (see  Ex.  3,  p.  92). 

It  is  further  assumed  that  the  diu'ivatives  u'.,.,  ?/^,  r\.,  v\^  are  continuous 
and  that  therefore  the  derivative  /''(".)  is  continuous.*  The  function 
is  then  said  to  be  an  tinaliiflc  funrfion  (§  126).  All  the  functions  of  a 
complex  variable  here  to  l)e  dealt  Avith  are  analytic  in  general,  although 
they  may  be  alloAved  to  fail  of  being  analytic  at  certain  specified  points 
called  sinr/ii/iir ]/oi')ifs.  The  adjective  "analytic"  may  therefore  usually 
be  omitted.    The  equations 

if  =/(-)     ('!■      "  =  "(•'■'  !/)^  >'  =  '■(•'■?  //) 

define  a  transformation  of  the  .'•//-] )lane  into  the  //r-])lane,  oi',  V)i'iefer,  of 
the  x-plane  into  the  vr-plane;  to  each  point  of  the  former  corresponds 
one  and  only  one  point  of  the  latter  (§  63).    If  the  Jacobian 

*  It  may  bo  jji-ovcd  tliat,  in  tlic  case  of  fuiictiims  of  a  (•omi)lex  vai'iablf,  tlu^ 
contiimity  of  the  (U'i'ivati\t'  follows  from  its  existeiii'e,  l)ut  the  proof  will  not  l)e 
yiveii  here. 

47«; 


COMPLEX  VARIABLE  477 

of  the  transformation  does  not  vanish  at  a  point  z^,  the  equations  may 
be  solved  in  the  neighborhood  of  that  poinly,  and  hence  to  each  point 
of  the  second  plane  corresponds  only  one  of  the  first: 

x  =  x(u,r),  i/  =  i/(u,v)     or     ^  =  0(m-). 

Therefore  it  is  seen  that  if  ic  =  f(z)  is  analytic  in  the  neiglihorJiood 
of  z  =  z^,  and  if  the  dei'icativef'(z^)  does  not  vanish,  the  function  may  he 
solved  as  z  =  (f>(i''),  where  ^  is  the  inverse  function  of  /,  and  is  like- 
wise analytic  in  the  neighborhood  of  the  point  w  —  n-^.  It  may  readily 
be  shown  that,  as  in  the  case  of  real  functions,  the  derivatives  /'("-)  and 
4>'(t''')  ^i"6  reciprocals.  Moreover,  it  may  be  seen  that  the  transfor mo- 
tion is  confonnal,  that  is,  that  the  angle  between  any  two  curves  is 
unchanged  by  the  transformation  (§  63).    For  consider  the  increments 

As  Az  and  Aw  are  the  chords  of  the  curves  before  and  after  transforma- 
tion, the  geometrical  interpretation  of  the  equation,  apart  from  the  infin- 
itesimal ^,  is  that  the  chords  Az  are  magnified  in  the  ratio  \f'(z^)\  to  1 
and  turned  through  the  angle  oi  f'(z^)  to  obtain  the  chords  Air  (§  72). 
In  the  limit  it  follows  that  the  tangents  to  the  ^r-curves  are  inclined  at 
an  angle  equal  to  the  angle  of  the  corresponding  ^.-curves  plus  the  angle 
of /'(.tg).    The  angle  between  two  curves  is  therefore  unchanged. 

The  existence  of  an  inverse  function  and  of  the  geometric  interpre- 
tation of  the  transformation  as  conformal  both  become  illusory  at  points 
for  which  the  derivative /'(.r)  vanishes.  Points  Avhere /' (,^)  =  0  are 
called  cvitical points  of  the  function  (§  183). 

It  has  further  been  seen  that  the  integral  of  a  function  wliich  is  ana- 
lytic over  any  simply  connected  region  is  independent  of  the  path  and 
is  zero  around  any  closed  path  (§  124) ;  if  the  region  be  not  simply  con- 
nected but  the  function  is  analytic,  the  integral  about  any  closed  path 
which  may  be  shrunk  to  nothing  is  zero  and  the  integrals  about  any 
two  closed  paths  which  may  be  shrunk  into  each  other  are  equal  (§  125). 
Purthermore  Cauchy's  result  that  the  value 


-^  ^  ^       2  7ri        t~z 

Jo 


dt  (3) 


of  a  function,  which  is  analytic  upon  and  within  a  closed  path,  may  be 
found  by  integration  around  the  path  has  been  derived  (§  126).  By  a 
transformation  the  Taylor  development  of  the  function  has  been  found 
whether  in  the  finite  form  with  a  remainder  (§  126)  or  as  an  infinite 
series  (§  167).     It  has  also  been  seen  that  any  infinite  power  series 


478  THEORY   OF   FUNCTIONS 

which  converges  is  differentiable  and  hence  defines  an  analytic  function 
■within  its  circle  of  convei'gence  (§  166). 

It  has  also  been  shown  that  the  sum,  difference,  product,  and  quotient 
of  any  two  functions  will  be  analytic  for  all  points  at  which  Ijoth  func- 
tions are  analytic,  except  at  the  points  at  which  the  denominator,  in  the 
case  of  a  quotient,  may  vanish  (Ex.  9,  p.  163).  ■  The  result  is  evidently 
extensible  to  the  case  of  any  rational  function  of  any  number  of  analytic 
functions. 

From  the  possibility  of  development  in  series  follows  that  {/  ttro 
functions  (f re  analytic  in  the  neirjlthorlinod  of  a  point  and  liai-e  identical 
values  vpon  anij  curve  draa-n  throurjli  that  point,  or  even  U])on  any  set 
of  points  which  approach  that  point  as  a  limit,  then  tlie  functions  are 
identicallij  equal  witliin  their  common  circle  of  eonvercjence  and  over  all 
regio7is  rvhich  can  he  reached  hij  (§  169)  continidng  tit e  functions  anali/ti- 
calbj.  The  reason  is  that  a  set  of  points  converging  to  a  limiting  point 
is  all  that  is  needed  to  prove  that  two  power  series  are  identical  |)ro- 
vided  they  have  identical  values  over  the  set  of  points  (Ex.  9,  \).  439). 
This  theorem  is  of  great  importance  because  it  shows  that  if  a  function 
is  defined  for  a  dense  set  of  real  values,  any  one  extension  of  the  defi- 
nition, which  yields  a  function  that  is  analytic,  for  those  values  and  for 
complex  values  in  their  vicinity,  must  be  equivalent  to  any  other  such 
extension.  It  is  also  useful  in  discussing  the  principle  of  permanence  of 
form  :  for  if  the  two  sides  of  an  equation  are  identical  for  a  set  of 
values  which  possess  a  point  of  condensation,  say.  for  all  real  rational 
values  in  a  given  interval,  and  if  each  side  is  an  analytic  function,  then 
the  e(|uation  must  ])e  true  for  all  values  which  may  be  reached  by  ana- 
lytic continuation. 

For  example,  the  e(iuation  sin  ,r  —  cnsi^  tt  —  ,'•)  is  known  to  liold  for  the  vahies 
0  =  ,r  =  I  TV.  Moreover  the  functions  sin  z  and  cos  z  are  analytic  for  all  values  of  z 
whether  the  detinition  l;)e  u'iven  as  in  §  74  or  whether  the  functions  be  considered 
as  defined  Ijy  their  jtower  series.  Hence  the  equation  nuist  liold  for  all  real  or 
complex  values  of  /.  In  like  manner  from  the  eipiation  co'  =  v^-n  whieh  holds 
for  real  rational  ^'Xponents.  the  e(iuation  t~e"'  —  c~  +  "'  holdiiiij,-  for  all  i-eal  ami  im- 
a^'inary  exponents  may  he  deduced.  For  if  //  he  <,dven  any  rational  value,  the 
functions  of  .c  <m  each  side  (jf  the  sii^n  are  analytic  for  all  \alues  of  ,r  real  or  com- 
plex, as  may  he  i^^'i'n  most  easily  by  consideriuir  the  exponenlial  as  delined  by  its 
l)Ower  series.  Hence  tlie  e(juation  holds  when  x  has  any  complex  value.  Next 
consider  .f  as  fixed  at  any  desired  complex  value  ami  h-t  tlie  two  sides  be  con- 
sidered as  functions  of  //  regarded  as  complex.  Jt  f(.illows  that  the  enuatiou  nuist 
hold  for  any  value  of  //.    The  CMiuation  is  therefore  true  for  any  value  of  z  and  v. 

179.  Suppose  that  a  function  is  analytic  in  all  ]i()ints  of  a  region  ex- 
ce})t  at  some  oiu'  point  within  tlie  region,  and  let  it  be  assumed  that 


COMPLEX  VAEIABLE 


479 


the  function  ceases  to  be  analytic  at  that  point  because  it  ceases  to  be 
continuous.  The  discontinuity  may  be  either  finite  or  infinite.  In  case 
the  discontinuity  is  finite  h>t  \f(z)\<  G  in  the  neighborhood  of  the 
point  '.'  =  (I  of  discontinuity.  Cut  the  point  out 
with  a  small  circle  and  apply  Cauchy's  Integral  to 
a  ring  surrounding  the  point.  The  integral  is  appli- 
cable because  at  all  points  on  and  within  the  ring 
the  function  is  analytic.  If  the  small  circle  be 
replaced  by  a  smaller  circle  into  which  it  may  be 
shrunk,  the  value  of  the  integral  will  not  be  changed. 


A^ 


is-rs 


1   2 


Xow  the  integral  about  y,-  Avhich  is  constant  can  be  made  as  small 
as  desired  l)y  taking  the  circle  small  enough;  for  \f{^)\<.  G  and 
\t  —  ~~\^\o  —  z\  —  7';,  where  r,-  is  the  radius  of  the  cii'cle  y,-  and  hence 
the  integral  is  less  than  2  7ry/''/[|,v  —  r/ 1  —  r,-].  As  the  integral  is  con- 
stant, it  must  therefore  be  0  and  may  be  omitted.  The  remaining  inte- 
gral about  (\  however,  defines  a  function  Avhich  is  analytic  at  z  =  a. 
Hence  if  /"(")  be  chosen  as  defined  by  this  integral  instead  of  the 
original  definition,  the  discontinuity  disappears.  Finite  disrontinuifirs 
niai/  f]i('i'>'f())-('  he  considered  as  due  to  bad  judgment  in  defining  a 
function  (it  some  point  ■  and  may  therefore  be  disregarded. 

In  the  case  of  infinite  discontinuities,  the  function  may  eitlier  become 
infinite  for  oil  met//o(/s  of  (ijijiroaeJi  to  the  point  of  discontiiniity,  or  it 
may  become  infnite  for  some  metJiods  of  approueh,  and.  reuia in  finite  for 
otiii'r  luet/iods.  In  the  first  case  the  function  is  said  to  have  a  jjole  at 
the  ])oint  -'  =  a  of  discontinuity;  in  the  second  case  it  is  said  to  have 
an  essential  singuiariti/.  In  the  case  of  a  pole  consider  the  reciprocal 
function 


F(z)  = 


m 


z  ^  a,  F{a)  =  0. 


The  function  F(z)  is  analytic  at  all  points  near  z  =  n.  and  remains 
finite,  in  fact  approaches  0,  as  z  approaches  a.  As  F{a')  =  0,  it  is  seen 
tliat  /•'(■-)  ^^'^^  ^^'^  finite  discontini;ity  at  «  =  a  and  is  analytic  also  at 
z  =  a.    Hence  the  Taylor  ex})ansion 

F{^  =  a,lz  -  ay-  4-  a^,^^^{z  -  af-^^  -f  •  ■  • 


is  pro])er.    If  E  denotes  a  function  neither  zero  nor  infinite  at  z  =  a, 
the  followimj-  transformations  mav  be  made. 


480  THEORY  OF  FUNCTIONS 

f(rA  =  _J---i^  +       ^'—'^^       +■■■  +  -^^ 

In  other  words,  a  function  which  has  a  pole  at  x  =  ^  inay  l)e  Avritten 
as  the  product  of  some  i)Ower  iz  —  r/)"^'"  by  an  /^-function;  and  as  the 
7s-function  may  be  expanded,  the  function  may  be  expanded  into  a 
power  series  which  contains  a  certain  number  of  negative  powers  of 
{z  —  d').  The  ardor  m  of  the  highest  ncr/atlre  power  is  called  tlie  order 
of  the  pole.    Compare  Ex.  5,  p.  449. 

If  the  function /'(,-.')  be  integrated  around  a  closed  curve  lying  within 
the  circle  of  convergence  of  the  series  C^  +  C ^{z  —  ")  +  ■••,  then 

r  r  r     dz  r  r  ,dz 

Jo 

or  f  fiz)dz  =  2  7rlC_,;  (4) 

Jo 


for  the  first  m  —  1  terms  may  be  integrated  and  vanish,  the  term 
C_i/(,v  —  a)  leads  to  the  logarithm  6'_i  log  (.■s  —  r/)  which  is  multiple 
valued  and  takes  on  the  increment  2irlC_-^,  and  the  last  term  vanishes 
because  it  is  the  integral  of  an  analytic  function.  The  total  value  of 
the  integral  of  f(z)  about  a  small  circuit  surrounding  a  pole  is  there- 
fore 2  7rlC_^.  The  vahie  of  the  integral  about  any  larger  circuit  Avithin 
which  tlie  function  is  analytic  except  at  z  =  a  and  which  may  be  shrunk 
into  the  small  circuit,  will  also  be  the  same  (piantity.  The  coefficient 
C_i  of  the  term  (,^  —  ff)~^  is  called  the  residue  of  the  jjo/e  ;  it  cannot 
vanish  if  the  pole  is  of  the  first  order,  but  may  if  the  ])ole  is  of  higher 
order. 

The  discussion  of  the  behavior  of  a  function  f(z)  when  ,-.'  becomes 
infinite  may  be  carried  on  by  making  a  transformation.    Let 

z'  =  l,         z  =  l,        f(z)=f(^  =  F(z').  (5) 


To  large  values  of  z  cori'espond  small  values  of  ,-.;';  if  f(z)  is  analytic 
for  all  large  values  of  z,  then  F(z')  will  be  analytic  for  values  of  z'  near 
the  origin.  At  z'  =  0  the  function  F(z')  may  not  be  defined  T)y  (5)  ;  but 
if  F(z')  remains  finite  for  small  values  of  z',  a  definition  may  be  given 
so  that  it  is  analytic  also  at  z'  —  0.    In  this  case  F(0)  is  said  to  be  the 


CO:\IPLEX  VARIABLE  481 

value  of  f(z)  when  z  is  infinite  and  the  notation  /(o))  =  /"(O)  may 
be  used.  If  F(z')  does  not  remain  finite  but  has  a  poh'  at  x'  =  0,  tlien 
f{z)  is  said  to  have  a  pole  of  the  same  order  at  .-.'  =  oc;  and  if  F{z^) 
has  an  essential  singularity  at  ':'  =  0,  then  /(.--')  is  said  to  have  an  essen- 
tial singidarity  at  2;  =  od.  Clearly  if  f{z)  has  a  pole  at  ,v  =  a>,  the  value 
of  /"(-)  must  become  indefinitely  great  no  matter  how  z  becomes  infi- 
nite; but  if  /('-')  has  an  essential  singularity  at  ?;  =  x,  there  will  be 
some  ways  in  which  z  may  become  infinite  so  that  f(z)  remains  finite, 
while  there  are  other  ways  so  that  f(z)  becomes  iiifinite. 

Strictly  speaking  there  is  no  point  of  the  ?.'-plane  which  corresponds 
to  z'  =  0.  Nevertheless  it  is  convenient  to  speak  as  if  there  were  such 
a  point,  to  call  it  tJie  jmbit  at  infinit)/,  and  to  designate  it  as  z  =  oc.  If 
then  F(z')  is  analytic  for  ^'  =  0  so  that  /'(••)  ^^''^J  ^^  ^^^^  ^^  ^6  analytic 
at  infinity,  the  expansions 

F{z')  =C,+  C^  -f  C^^  +  •  •  ■  -f  C,,^'"  -f-  •  •  •  = 

(\      r,,  ( ' 

are  valid  ;  the  function  ,/'(-')  ^^''^^  been  cx/xindcil  ahoiit  the  jioint  at  lnp'?i- 
ity  Into  a  desr ending  power  series  in  z,  and  the  series  will  converge  for 
all  points  z  outside  a  circle  |.v]  =  U.    For  a  jmjIc  of  order  w  at  infinity 

/(^)  =  C_,„z'"  +  ('-,„, ^r:'"-'  +  . . .  +  C_,z  +  C^  +  ^  +  ^  +  •  •  • . 

Simply  because  it  is  convenient  to  introduce  tlie  conce])t  of  the  point 
at  infinity  for  the  reason  that  in  many  ways  the  totality  of  large  values 
for  z  does  not  differ  from  the  totality  of  values  in  the  neighborhood  of 
a  finite  point,  it  should  not  be  inferred  that  the  point  at  infinity  has 
all  the  pro})erties  of  finite  points. 

EXERCISES 

1.  Discuss  sin  (x  +  y)  =  sin  x  cos  y  +  vos  x  sin  y  for  permanence  of  form. 

2.  If  f{z)  lias  an  essential  singnlarity  at  z  =  «,  show  that  l/f{z)  has  an  essential 
sinp:ularity  at  z  =  a.  Hence  infer  that  there  is  some  method  of  approach  to  2  =  a 
such  that  f{z)  =  0. 

3.  By  treating  f{z)  —  c  and  [f{z)  —  c]-i  show  that  at  an  essential  singularity  a 
function  may  be  made  to  approach  any  assigned  value  c  by  a  suitable  method  of 

approaching  the  singular  point  z  =  n. 

4.  Find  the  order  of  the  poles  of  these  functions  at  the  origin  : 

(a)  cot  2,  (/3)  csc2rlng(l— 2),         (7)  2(sin  z  —  tan  z)-i. 


482 


THEOKY  OF  FUNCTIONS 


5.  Show  that  if /(z)  vanishes  at  z  =  a  once  or  n  times,  the  <jnotient  f'{z)/f{z)  lias 
the  residue  1  or  n.  Show  that  if  f{z)  has  a  i)ole  of  tlie  ?ntii  order  at  z  =  «,  tiie 
(juotient  has  the  residue  —  m. 

6.  From  Ex.  5  prove  tlie  important  tlieorem  that :  If  f{z)  is  analytic  and  does 
not  vanish  upon  a  closed  curve  and  has  no  siniiidarities  otiier  than  poles  within 
the  ci;rve,  then 


—  r 


.nz) 

o  f{z) 


dz  =  ?(j  +  v.,  + 


7/A-  —  '",   —  111; 


mi  —  N  —  M, 


where  N  is  the  total  miml)er  of  roots  of  f{z)  =  0  within  the  curve  and  M  is  the 
sum  of  the  orders  of  the  i)oles. 

7.  Apxjly  Ex.  0  to  \/F{z)  to  show  that  a  polynomial  P {z)  of  the  ?ith  order  has 
just  n  roots  within  a  sufficiently  lari^e  curve. 

8.  Trove  that  e«  caimot  vanish  for  any  finite  value  of  z. 

9.  Consider  the  residue  of  zf'{z)/f{z)  at  a  ])()!('  or  vanishing,''  point  of /(z).  In 
I)articular  prove  that  if /(z)  is  analytic  and  does  not  vanish  upon  a  closed  curve 
and  has  no  sint^ularities  but  poles  within  the  curve,  then 

]       />  zf'(A 

■- — :        -^-r—  dz  =  riAU  +  «.,«,  +  •  •  •  +  iU"/[-  —  "'i''i  -  mj)., ?/*;/;,, 

2  7rt  Jq  f{z) 

where  a^,  a,,  •  •  • ,  a^.  and  ?;j,  r/o,  ■  ■  ■  ^  rik  are  the  positions  and  orders  of  the  roots, 
and  />j,  />.,,  •  •  • ,  hi  and  ///,.  »/.,,  ■  ■  • ,  rni  of  the  poles  of /(z). 

10.  Trove  that  OjCz).  p.  4()!»,  has  only  one  root  within  a  rectanii'le  2  /v  by  2  IK'. 

11.  State  the  beha\ioi'  (analytic,  pole,  or  essential  sMiuularity)  at  z  =  oo  for  : 

(a)  z^  +  2z.         (/3)  r>',         (7)  z/(l  +  z),  (5)  z/{z^  +  1). 

12.  Show  that  if /(z)  =  (z  -  aY-E{z)  witli  -  1  <  A-  <  0,  the  inte.ural  of /(z)  about 
an  infinitesimal  t'ontour  sui'roundin^-  z  =  rr  is  infinitesimal.  What  analogous  theo- 
rem holds  foi'  an  infinite  contour  '? 

180.  Characterization  of  some  functions.  The  study  of  the  limita- 
tions whicli  arc  ])tit  ii])oii  a  fum-tioii  when  certain  of  its  properties  are 
known  is  important.  For  example,  n  fiincflon  irhlch  is  (indh/fir.  for  (ill 
raliti's  of  z  hicl iid'ind  filso  z  =  co  is  <i  consfunf.  To  sliow  this,  note  that 
as  the  ftmction  nowlicre  liccomes  infinite,  [./'(■"-■)|  <  '^•-  Consider  tlie  dif- 
ference/'('•„)  —  ,/'(**)  hctwecn  the  value  at  any  ])oint  z  =  z^^  and  at  the 
origin.  Take  a.  circle  conceiiti'ic  with  z  =  0  and  of  radius  Jl  >  |,-.'^|. 
Then  by  Caucliy's   Inteoral 

//^  )  _  /Y0)|<  ^^ '    '■' 


A^d-fm 


R  -I.v„ 


By  taking   /'  large  enough  the  difference,  which  is  constant,  may  l)e 
made  as  small  as  desired  and  henc*;  must  he  zero;   henc(>  f(z)  —  f(0). 


COiAIPLEX  VARIABLE  483 

Any  rational  function /(,-)  =  P(s)/(2(.t),  where  P{z)  and  Q{z)  are 
polynomials  in  z  and  may  be  assumed  to  be  devoid  of  common  factors, 
can  have  as  singularities  merely  poles.  There  will  be  a  pole  at  each 
point  at  which  the  denominator  vanishes;  and  if  the  degree  of  the 
numerator  exceeds  that  of  the  denominator,  there  will  be  a  pole  at  in- 
finity of  order  equal  to  the  difference  of  those  degrees.  Conversely  it 
may  be  shown  that  any  function  u-h'ich  has  no  other  singularity  than  a 
pole  of  the  mtli  order  at  infinity  must  he  a  jjolynomial  of  the  vitJi  order ; 
that  if  tit  e  only  singularities  are  a  finite  number  of  poles,  vhetlier  at  in- 
finity or  at  other  points,  the  function  is  a  rational  function ;  and  finally 
that  the  l^noxcledge  of  the  zeros  and  j)oles  u-ifh  the  multiplicity  or  order 
of  each  is  sufficient  to  determine  the  function  excepjt  for  a  constant 
multiplier. 

For,  ill  the  first  place,  if  f{z)  is  analytic  except  for  a  pole  of  the  mth  order  at 
infinity,  the  function  may  be  expanded  as 

f{z)  =  a-,„z"'  +  ■■■  +  u-iz  +  «y  +  a^z-1-  +  a.,z-"-  +  ■  •  • , 

or  f{z)  -  [r(_  ,„z"'  +  •  •  •  +  (t-iz]  =  a,,  +  a^z-'^  +  a.,z-"  +  •  •  • . 

Tlie  function  on  the  right  is  analytic  at  intinity,  and  so  must  its  equal  on  the  left 
be.  The  function  on  the  left  is  the  difference  of  a  function  which  is  analytic  for 
all  finite  values  of  z  and  a  polynomial  which  is  also  analytic  for  finite  values. 
Hence  the  function  on  the  left  or  its  equal  on  the  right  is  analytic  for  all  values 
of  z  including  z  =  x.,  and  is  a  constant,  namely  «^.    Hence 

f[z)  =  «|,  +  U-iz  +  ■  ■  ■  +"_,„z"'     is  a  polynomial  of  order  m. 

In  the  second  place  let  z^,  z.,.  ■  ■  ■ .  z/,-.  co  be  poles  of  f{z)  of  the  respective  orders 
m^,  m.,,  •  •  • ,  nik,  m.   The  function 

0  {z)  =  {z  -  z,)"'^z  -  z.f"-^  ■■■{z-  z,)"''^-f{z) 

will  then  have  no  singularity  but  a  pole  of  order  ni^  +  m„  +  •  •  •  +  '"a-  +  m 
at  infinity;  it  will  therefore  be  a  polynomial,  and  f{z)  is  rational.  As  the 
numerator  0(z)  of  the  fraction  cannot  vanish  at  z^,  z.,,  •••,  z>;..  but  must  have 
THj  +  ?n„  +  •  •  •  +  ?/u-  +  m  roots,  the  knowledge  of  these  roots  will  determine  the 
numerator  (p(z)  and  hence /(z)  except  for  a  constant  multiplier.  It  should  be 
noted  that  if  f{z)  has  not  a  pole  at  infinity  but  has  a  zero  of  order  m,  the  above 
reasoning  holds  on  changing  m  to  —  m. 

When  f(z)  has  a  ])ole  at  z  =  a  of  tlie  mt\\  order,  the  expansion  of 
/(-:)  about  the  pole  contains  <,'ertain  negative  powers 

p{z  -  a)  = —  + ~—^  H h  ■ 

^  ^        (,-;  —  (/)'"        (z  —  a)'"    ^  z  —  a 

and  the  difference  f{z)  —  P(z  —  a)  is  analytic  at  z  =  a.  The  terms 
P(z  —  a'j  are  called  the  principal  part  of  tlie  function  f{z)  at  the  pole  a. 


484  THEORY  OF  FUNCTIONS 

If  the  function  lias  only  a  finite  nnuil)ei'  of  finite  poles  and  the  prin- 
cipal parts  corresponding  to  each  pole  are  known, 

4>(z)  =f(^  -  P^  -  .J  -  7'i.  -z^ P^  -  z,) 

is  a  function  which  is  everywhere  analytic  for  finite  values  of  z  and 
behaves  at  .-s  =  oo  just  as  ./'(*-')  behaves  there,  since  ]\,  P„  •••,  P^. all 
vanish  at  z  —  cc.  If  /(*-')  is  analytic  at  z  =  cc,  then  <j>(ji)  is  a  constant; 
if  f(z)  has  a  pole  at  z  =  cc,  then  ^  (z)  is  a  polynomial  in  z  and  all  of 
the  polynomial  exce])t  the  constant  term  is  the  ])rincipal  part  of  the 
pole  at  infinity.  Hence  if  a  function  lias  no  singuhd-itles  except  a  finite 
number  of  ^Jolea,  ((ml  the  p't'incijxd  ixais  ((t  tltese  j^oles  are  knoivn,  tlie 
function  is  deter  oiined  except  for  an  (((hlitire  constdut. 

From  the  above  considerations  it  appears  that  if  a  function  has  no 
other  singularities  than  a  finite  number  of  i)oles,  the  function  is  ra- 
tional; and  that,  moreover,  the  function  is  determined  in  factored  form, 
exce2)t  for  a  constant  multi])lier,  when  the  positions  and  orders  of  the 
fiinte  poles  and  zei'os  are  known ;  or  is  determined,  except  for  an  addi- 
tive constant,  in  a  development  into  partial  fractions  if  the  positions 
and  principal  parts  of  the  ])()les  arc  known.  All  single  valued  functions 
other  than  rational  functions  luust  therfsfon;  have  either  an  infinite 
number  of  poles  or  some  essential  singularities. 

181.  The  ex})onential  function  e~  =  r''(cos  //  +  ^  sin  y)  has  no  finite 
singularities  and  its  singularity  at  infinity  is  necessarily  essential.  The 
function  is  periodic  (§  74)  with  the  period  2  iri,  and  hence  will  take  on 
all  tlie  different  valutas  Avhich  it  can  have,  if  z,  instead  of  being  allowed 
all  values,  is  restricted  to  have  its  pure  imagi- 
nary part  //  between  two  limits  (/^=  (/  <.  //^-f  27r; 

that  is,  to  consider  the  values  of  e~  it  is  merely     -'A 

necessary  to  consider  tlie  values  in  a  strip  of 
the  ,^-plane  ])arallel  to  tlie  axis  of  reals  and  of  breadth  2  tt  (but  lacking 
one  edge).  For  convenience  the  sti-ip  m;iy  be  taken  immediately  above 
the  axis  of  reals.  Tlu',  function  c~  becomes  infinite  as  ,-.;  moves  out 
toward  the  right,  and  zero  as  z  moves  out  towartl  the  left  in  the  strip, 
ir  c  =  (f.  -\-  fii  is  any  number  other  than  0,  there  is  one  and  only  one 
])oint  in  tlu;  stri])  at  which  e~  =  c.     For 

a  .    .         b 


.Z  +  iTTl 


=    V^r   +   0~         antl  cos    //    +    t    Sm    //   =    -  yrrr — -n \-    I 


have  only  one  solution  for  ,r  and  u\\\\  one  i'or  //  if  //  be  restricted  to  an 
interval  2  tt.  All  other  points  t'oi-  which  r-  =  c  have  the  same  value  for 
./•  and  some  value  //  ±  'Idir  Un-  y. 


COMPLEX  VARIABLE  485 

Any  rational  function  of  e'',  as 

will  also  have  the  period  2  iri  ^^'hen  x  moves  off  to  the  left  in  the 
strip,  R  (ff)  will  approach  Ca^/b^^^  if  ^^^  =^  0  and  will  become  infinite  if 
h^^  =  0.  AYhen  z  moves  off  to  the  right,  R  {(f)  must  become  infinite  if 
n  >  m,  approach  C  if  n  =  m,  and  approach  0  if  m.  <  vi.  The  denomi- 
nator may  be  factored  into  terms  of  the  form  («^  —  ay,  and  if  the  frac- 
tion is  in  its  lowest  terms  each  such  factor  will  represent  a  pole  of  the 
A-th  order  in  the  strip  because  e^  —  a  =  0  has  just  one  simple  root  in 
the  strip.  Conversely  it  may  be  shown  that:  An u  function  /(■:)  vlik-Ji 
lias  tlie  ijer'iod  2'Trl,  which  fiirtlicr  Itas  no  singularities  hut  a  Jinitc 
number  of  jjoles  In  each  strip,  and  irhlcJi  either  becomes  infnite  or  ap- 
proaches a  finite  limit  as  z  moves  off  to  the  rhjht  or  to  the  left,  must  be 
f(z)^  Rfe^^,  ((  rational  function  of  e^. 

The  proof  of  this  theorem  requires  several  steps.  Let  it  first  be  assumed  that/(2) 
remains  finite  at  tlie  ends  of  the  strip  and  lias  no  poles.  Then/(z)  is  finite  over  all 
values  of  z,  including  z  =  co,  and  nuist  be  merely  constant.  Xext  let  f{z)  remain 
finite  at  the  ends  of  the  strip  but  let  it  have  poles  at  some  points  in  the  strip.  It  will 
be  shown  that  a  rational  function  7i  (c^)  may  be  constructed  such  that  f{z)  —  R  (e^) 
remains  finite  all  over  the  strip,  including  the  portions  at  infinity,  and  that  there- 
fore f{z)  =  R{e^)  +  C.    For  let  the  principal  part  oi  f{z)  at  any  pole  z  =  c  be 

P(z  —  c)  = ^ 1 ^+ 1 + ;      then     * = h  •  •  • 

^  {z-  c)k      (2  _  c)^--i  z-c  (e^  -  e'^f      (z  -  c)^" 

is  a  rational  functit)n  of  e^  wliicli  i-cniains  finite  at  both  ends  of  the  strip  and  is 
such  that  the  difference  between  it  and  F {z  —  c)  ov  f{z)  has  a  pole  of  not  more 
than  the  {k  —  l)st  order  at  z  =  c.  By  subtracting  a  number  of  such  terms  from 
/(z)  the  pole  at  z  =  c  may  be  eliminated  without  introducing  any  new  pole. 
Thus  all  the  poles  may  be  eliminated,  and  the  result  is  proved. 

Next  consider  the  case  where /(z)  becomes  infinite  at  one  or  at  both  ends  of  the 
stri}).  If  /(z)  happens  to  approach  0  at  one  end,  consider  f{z)  +  (',  which  cannot 
approach  0  at  either  end  of  the  strip.  Now  if /(z)  ov  f{z)  +  C,  as  the  case  may  be, 
had  an  infinite  number  of  zeros  in  the  strip,  these  zeros  would  be  confined  within 
finite  limits  and  would  have  a  point  of  condensation  and  the  function  would  vanish 
identically.  It  must  therefore  be  that  the  functiuu  has  only  a  finite  number  of 
zeros;  its  reciprocal  will  therefore  have  only  a  finite  number  of  poles  in  the  strip 
and  will  remain  finite  at  the  ends  of  the  strips.  Hence  the  reciprocal  and  conse- 
quently the  function  itself  is  a  rational  function  of  e^.  The  theorem  is  completely 
demonstrated. 

If  the  relation  f(z  -\-  w)  —  f{z)  is  satisfied  by  a  function,  the  func- 
tion is  said  to  have  the  period  w.  The  function  /(2  Trlz/ui)  will  then 
have  the  jx'riod  2  tt/.  Hence  it  follows  that  Iff(z)  I/as  tJie  period  w, 
becomes  Infinite  or  ronalns  finite  at  tlie  ends  of  a  strip  of  rector  tireadth 


486 


THEORY   OF  FUNCTIONS 


<i),  and  uas  no  slngularltb's  hut  a  jinlfe  nuiiiln'r  of  poles  in  the  strip,  tlie 
function  i.s  a  rational  function  of  ,'-'"'=/•",  In  particular  if  the  period 
is  2  TT,  the  function  is  rational  in  e'^,  as  is  the 
case  with  sins;  and  cos  z;  and  if  the  period  is 
TT,  the  function  is  rational  in  c'-^-,  as  is  tan  z. 
It  thus  appears  that  the  single  valued  elemen- 
tary functions,  namely,  rational  functions,  and 
rational  functions  of  the  exponential  or  trigonometric  functions,  have 
simple  general  properties  which  are  characteristic  of  these  classes  of 
functions. 

182.   Sui)pose  a  function  /(.~)  has  two  independent  periods  so  that 

f(z  +  cu)  ==  fix),         /(,.  +  <.';  =  f(z). 

The  function  then  has  the  same  value  at  z  and  at  any  point  of  the 
form  z  +  ?y/to  -f  nw',  where  m  and  7i,  are  ])Ositive  or  negative  integers. 
The  function  takes  on  all  the  values  of  which  it  is  capable  in  a  parallel- 
ou:ram  constructed  on  the  vectors  w  and  w'.     Such  ^  , 

a  function  is  called  doufil//  jjcriodic.  As  the  values 
of  the  function  are  the  same  on  opposite  sides  of 
the  parallelogram,  only  two  sides  and  the  one  in- 
clndt^d  vertex  are  siqiposed  to  belong  to  the  ligiu'e. 
It  has  been  seen  that  some  doubly  periodic  func- 
tions exist  ($  1~~);  but  without  reference  to  these 
special  functions  many  important  theorems  concerning  doubly  periodic, 
functions  may  be  proved,  subject  to  a  subsecjuent  demonstratitju  that 
the  functions  do  exist. 

If  a  d(jtihlij  periodic  function  Imx  rin  si/);/u/ti/-!tirs  in  tin-  pdriillfliiijririii, 
it  uiust  hi'  cirnsfant :  for  the  function  will  then  ha\e  no  singularities  at 
all.  Jfficit  pcriinlic  functinns  Imre  tlw  sn im-  j^crimls  and  lairi-  fli,'  seme 
poh-s  and  zeros,  (each  to  the  same  ordei-)  in  the  jxt rdlhlnfjrn m .  tin'  ijitu- 

tient  if  tJie  fii itetidHS   is  "   enUstiiut:    iftliei/  hiire   the  sniiie  pules   mid   tjie. 

same  prineipiil  pii  rts  nt  the  j,ides,  tlieir  diffennee  is  n  enustnnf.  In  these 
theorems  (and  all  those  following")  it  is  assumed  that  the  functions 
have  no  essential  singularity  in  the  jiarallelogram.  The  })roof  of  the 
theorems  is  left  to  the  reader.  Xi  f(z)  is  doulily  ]  )er  iodic, /"( ,-.)  is  als(j 
doulily  periodic.  Tlie  integral  of  a  doubly  ])eri()dic  fuiictifui  taken 
around  any  ])arallelogram  e(pial  and  jiarallel  to  tlu*  parallelogi'am  of 
periods  is  zero;  for  tlie  function  ]-e}ieats  itself  on  f)p}iosite  sides  of  the 
figure  while  the  differential  '/,•.■  changes  sign.    Hence  in  particular 


X- 


f{z)dzr^\), 


r  f(z) 


dz  =  0, 


'O 


f(r:)dz 


=  0. 


COMPLEX  VARIABLE  487 

The  first  integral  shows  tliat  the  sum  of  tJie  residues  of  the  poles  hi  tlie 
Ijarallelorjram  is  zero  ;  tlie  second,  that  the  numher  of  zeros  is  equal  to 
the  numher  of  poles  provided  multiplicities  are  taken  into  account;  the 
third,  that  the  numher  of  zeros  of  f(z)  —  C  is  the  same  as  tlie  numher  of 
zeros  or  px)les  off(z),  because  the  poles  oif(z)  and/(s;)  —  C  are  the  same. 
The  common  number  m  of  poles  of  /(.")  or  of  zeros  of  f(z)  or  of  roots 
of  /(.?)  =  C  in  any  one  parallelogram  is  called  the  order  of  the  doxdAy 
period ic  function.  As  the  sum  of  the  residues  vanishes,  it  is  impossible 
that  there  should  be  a  single  pole  of  the  first  order  in  the  parallelogram. 
Hence  there  can  be  no  functions  of  the  first  order  and  the  simplest 
possible  functions  Avould  be  of  the  second  order  with  the  expansions 

7 77,  +  c  +  cfz  -c)-\ or h  c^  H and \-  c'  -\ 

in  the  neighborhood  of  a  single  pole  at  z  =  a  of  the  second  order  or  of 
the  two  poles  of  the  first  order  at  z  —  a^  and  -.;  =  a,^  Let  it  be  assumed 
that  when  the  periods  w.  w'  are  given,  a  doubly  periodic  function  (/(.", «) 
with  these  periods  and  witli  a  double  pole  at  z  =  a  exists,  and  similarly 
that  /;  (.?,  «p  «.-,)  with  simple  ])oles  at  a^  and  o.,  exists. 

Any  doahhj  periodic  function  f{z)  vitlt  tlie  periods  u>,  w'  niay  he  ex- 
pressed as  a  jJoli/noniial  in  the  functions  [/(z,  a)  and  h  (z,  a^,  r/.,)  of  the 
second  order.  For  in  the  first  place  if  the  function  f{:S)  lias  a  pole  of 
even  order  2 /.•  at  z  =  a,  then  f(z)  —  C[f/(.v,  a)f,  Avhere  C  is  properly 
chosen,  will  have  a  pole  of  order  less  than  2  ]:  at  z  =  a  and  will  have 
no  other  poles  than  f(z).  Hence  the  order  of  /'(■-)  —  '"['/(■^'j  ''0]^  ^^  ^^ss 
than  that  of /(,v).  And  if /"(■•)  ^^^^  '"^  P*^^^*^'  ^^^  '^^^  order  2  /.•  +  1  at  z  =  a, 
the  function /(,?)  —  ('[;/(z,  a)f]((z,  a,  J>),  with  the  proper  choice  of  C, 
will  have  a  pole  of  order  2  h  or  less  at  -.■  =  a.  and  will  gain  a  simple 
pole  at  z  =  />.  Thus  although  /'  —  Cf/'li  will  generally  not  be  of  lower 
order  than  /',  it  will  have  a  comY)lex  pole  of  odd  order  si)Iit  into  a  pole 
of  even  order  and  a  pole  of  the  first  order ;  the  order  of  the  former 
may  be  reduced  as  before  and  pairs  of  the  latter  may  be  removed.  By 
repeated  applications  of  the  process  a  function  may  be  obtained  which 
has  no  poles  and  must  be  constant.    The  theorem  is  therefore  proved. 

AVith  the  aid  of  series  it  is  possible  to  write  down  some  doubly  peri- 
odic functions.    In  particular  consider  the  series 


p<^  =  i  +  X 


(Z  —   UIW   —  llio'f  (^tllOJ  +  7lw'f 

and  7>'(.^)  =  -2Tt: 


(6) 


(.V   —   ///W  —  7lU)  ) 


488  THEOKY  OF  FUXCTIONS 

Avhere  the  second  2  denotes  summation  extended  over  all  values  of 
m,  n,  whether  i)Ositive  or  negative  or  zero,  and  2'  denotes  summation 
extended  over  all  these  values  exee})t  the  pair  ///  =  ?i  =  0.  As  the  sum- 
mations extend  over  all  })ossible  values  for  m,  n,  the  series  construeted 
for  z  -\-  i»  and  for  z  +  w'  must  have  the  same  terms  as  those  for  ,-.■,  the 
only  difference  being  a  different  arrangement  of  the  terms.  If,  there- 
fore, the  series  are  absolutely  convergent  so  that  the  order  of  the  terms 
is  immaterial,  the  functions  must  have  the  periods  w,  w'. 

Consider  first  tlie  convergence  of  the  series  p'{z).    For  z  =  moj  +  nw',  that  is,  at 
tlie  vertices  of  the  net  of  paraUelogranis  one  term  of  the  series  becomes  infinite 
and  tlie  series  cannot  converu'e.    But  if  z  be  restricted  to  a  finite  region  7t  about 
z  =  0,  tliere  ^vill  be  only  a  finite  number  of  terms 
which  can  become  infinite.    Let  a  parallelogram  P      -    -  /^"^ 

large  enough  to  surround  the  region  be  drawn,  and 
consider  only  the  vertices  which  lie  outside  this  par- 
allelogram. For  convenience  of  computation  let  the 
points  z  =  mw  +  nw'  outside  P  be  considered  as  ar- 
ranged on  successive  parallelograms  P.^,  P„,  •  •  • , 
Pi:  ■  •  •  .  If  the  number  of  vertices  on  P  be  v,  the 
number  on  P^  is  v  +  8  and  on  P/^.  is  v  +  8k.  The 
shortest  vector  z  —  mcj  —  no'  from  z  to  anj'  vertex  of  Pj  is  longer  than  a,  where 
a  is  the  least  altitude  of  the  parallelogram  of  periods.  The  total  contribution  of 
P^  to  iy{z)  is  therefore  less  than  {v  +  8)(t-°  and  the  value  contributed  by  all  the 
vertices  on  .successive  parallelograms  will  be  less  than 

,      v+8      y+8-2      v+  8-?>  i>+  8-k 

This  series  of  positive  terms  converges,  llenci'  the  infinite  series  for  p'{z),  when 
the  fii'st  terms  corresponding  to  the  vertices  within  ]\  are  disregarded,  converges 
ab.solutely  and  even  uniformly  so  that  it  represents  an  analytic  function.  The 
whole  .series  for  p'{z)  therefore  represents  a  doubly  periodic  function  of  the  tliird 
order  analytic  everywhere  except  at  the  vertices  of  the  parallelograms  where  it 
has  a  pole  of  tlie  third  order.  As  the  part  of  the  series  p'{z)  contributed  by  ver- 
tices outside  J'  is  uniformly  convergent,  it  may  be  integrated  from  0  to  z  to  give 
the  corresponding  terms  in  p(z)  which  will  also  be  ab.solutely  coiivergent  because 
the  terms,  groupeil  as  for  p'(z),  will  be  less  than  the  terms  of  IS  where  I  is  the 
length  of  the  ])atli  of  integration  from  0  to  z.  The  other  terms  of  p'{z).  thus  far 
disregarded,  may  lie  integrated  at  sight  to  obtain  the  eorresiionding  terms  i)i  2){z). 
Hence  p'{z)  is  really  the  derivative  oi  p  (z)  ;  and  as  p  (z)  converges  absolutely  ex- 
cept for  the  vertices  of  the  parallelograms,  it  is  clearly  doubly  periodic  oi  the 
Heco}id  order  with  the  periods  oi,  w',  for  the  same  reason  thatp'(2:)  is  periodic. 

It  has  therefore  ln'on  sliown  that  doubly  periodic  functions  exist, 
and  hence  the  tlicoicms  deduced  for  such  functions  are  valid.  Some 
further  important  theorems  are  indicated  among  the  exercises.  Tliey 
lead  to  the  inference  tliat  ;uiy  doublv  periodic  function  whieli  lias  the 


COMPLEX  VARIABLE  489 

periods  w,  w'  and  has  no  other  singularities  tlian  poles  may  be  expressed 
as  a  rational  function  of  ^y(,t:)  and  p'(z),  or  as  an  irrational  function  of 
2>(s:)  alone,  the  only  irrationalities  being  square  roots.  Thus  by  em- 
ploying only  the  general  methods  of  the  theory  of  functions  of  a 
complex  variable  an  entirely  new  category  of  functions  has  been  char- 
acterized and  its  essential  properties  have  been  proved. 

EXERCISES 

1.  Find  the  principal  parts  at  z  =  0  for  the  functions  of  Ex.  4,  p.  481. 

2.  PiTive  b}'  Ex.  (),  p.  482,  that  e~  —  c  =  0  has  only  one  root  in  tlie  strip. 

3.  How  does  e<''^>  behave  as  z  becomes  infinite  in  tlie  strip? 

4.  If  the  vahies  A'(t')  approaclies  when  z  becomes  infinite  in  tlie  strip  are  called 
exceptional  values,  show  that  Ii{c~)  takes  on  every  value  otlu^r  than  the  excep- 
tional values  k  times  in  the  strip,  k  beini;  the  greater  of  the  two  luimljers  77.  ?/(. 

5.  Show  b}'  Ex.  9,  p.  482,  that  in  any  parallelogram  oi  periods  the  sum  of  the 
positions  of  the  roots  less  tlie  sum  of  the  jiositions  of  the  poles  of  a  doubly  peri- 
odic function  is  mu)  +  nw',  where  m  and  J7  are  integers. 

6.  Show  that  the  terms  of  p'{z)  may  be  associateil  in  such  a  way  as  to  prove 
that_p'(— r)  =  —  p'{z),  and  hence  infer  that  the  expansions  are 

p\z)  =  —  2^-3  -I-  )>c^z  4-  ■ic.z^  +  ■  •  • ,         "idy  odd  powers, 

and  p{z)  —  Z--  -\-  CjZ-  -\-  c.,z*  +  ■  •  • ,         "idy  even  powers. 

7.  Examine  the  series  (0)  f or ^'(z)  to  sliow  that p'{l  w)  =2>'(.l  w')  =p'(l  w  -|-  ^  w')  =0. 
"Why  can  p'(z)  not  vanish  for  any  other  points  in  tlie  ]>arallelogram  ? 

8.  Let  p(l  o))  =:  r.  p{l  uj')  =  (,'.  p{l  w  +  I  w')  =  r".  Prove  tlie  identity  of  the 
doubly  periodic  functions  [p'(~)]"  '^'^"^^  '^[P  (-)  ~  ^']  [  /'  (~)  ~  ^'l  IP  (-)  ""  *-'']• 

9.  B}^  examining  the  series  dehning  p(z)  show  tliat  any  two  poiiUs  z  =  a  and 
z  =  a'  such  tliatp(rf)  =p(((')  are  symmetrically  situated  in  the  parallelogram  with 
respect  to  the  center  z  =  l{w  +  ui').    How  could  this  be  inferred  from  Ex.  5 '.' 

10.  AVith  the  notations  <j{z,  d)  and  ]i{z,  a^,  «.,)  of  the  text  show: 

im+jfUT)  ^  ^  ^^  p:(z)_+iyia)  ^  _  ^  ^^ 

piz)-p{a)  V  ^         p^r)^p^a) 

(^)  ]yiz)  +  p'(a^  _  P:(z)  +  p'ia^  ^  ,  ^^  ^^^        ^^^^^ 
p{z)-p{(i.-,)        p{z)-p{a^) 

1  Vp'(z)  +  p'{a)l  ■^_  ^  ^         _ 

^"  ilp(z)-p{a)]      ^w     ./v  ,    ;     i\         ;t 

■ilpiz)-p{")  J 

11.  Demonstrate  the  linal  theorem  of  the  text  of  ^182. 


490 


THEORY   OF   FLXCTIOXS 


12.  By  coiiibiiiiiifr  the  power  series  forp(2)  and  l>'{z)  show 

[p'{z)]-  -  4  [p(z)]3  +  •20c^p{z)  +  28r,,  =  Az-  +  higlier  powers. 
Hence  infer  that  the  right-hand  side  must  be  identically  zero. 

13.  Combine  Ex.  12  witli  Ex.  8  to  prove  e  +  e'  +  e"  =  0. 


14.  "With  the  notations  g.,  =  20  r^  and  <j.^  =  28  c.,  show 

dp 


p'{z)  =  V4  pHz)  -  y,p(~)  -  <li     or 


dz. 


d 


I  p  {z)dz.  show  that 


15.  If   f(2)   be  defined   bv t{z)=p{z)    or   ^{z) 

dz 
i'{z  +  oj)  —  ^"(2)  and  (;{z  +  w')  —  i'{z)  must  be  merely  constants  77  and  rj'. 

183.  Conformal  representation.    The  transfonnation  (§  178) 

ic  =/('i)      or      u  +  if  =  i>(.r,  11)  +  Irix.  I/) 

is  oonformal  between  the  planes  of  ,-:  and  u-  at  all  points  z  at  which 
/'(,-.•)  ^  0.  The  correspondence  between  the  planes  may  be  represented 
In*  ruling  the  ?;-plane  and  drawing  the  corresponding  rulings  in  the 
?/--plane.  If  in  particular  the  rulings  in  the  .v -plane  be  the  lines  ./•  =  const., 
_//  =  const.,  parallel  to  the  axes,  those  in  the  «*-plane  must  be  two  sets 
of  curves  which  are  also  orthogonal;  in  like  manner  if  the  ,v-i)lane  be 
ruled  V)y  circles  concentric  with  the  origin  and  rays  issuing  from  the 
origin,  the  w-plane  must  also  be  ruled  orthogonally  :  for  in  liuth  cases 
the  angles  between  curves  must  l)e  preserved.  It  is  usually  most 
convenient  to  consider  the  /'/•-plane  as  ruled  Avith  the  lines  11  =  const., 
r  =  const.,  and  hence  to  have  a  set  of  rulings  u(:i\  if)  =  c^,  rf.r,  if)  =  r,, 
in  the  '.--plane.  The  figiu-es  represent  several  different  cases  arising  from 
the  functions  • 


xc-plane       (1)       z—jyiaiie 

(1)   w  =  az  =  (,f^  +  If/)  (x  -f  li/) 


0        1 

- 

1 

> 

lOLT  ."  =  loLf  V./'- 


aan-> 


Consider  -?/•  =  ,-:'-',  and  apply  ])olar  cdrjrdinatc.; 
ir  =  J'  (ros  <J>  +  /  sin  <t>)  =  /•-{Cos  2  (^  +  /  sin  1' 


COMPLEX  VARIABLE  491 

To  any  point  (r,  <^)  in  the  2:-plane  corresponds  (A'  =  /•-,  4>  =  2  </>)  in  the 
(T-plane  ;  circles  about  z  =  0  become  circles  about  w  =  0  and  rays  is- 
suing from  z  =  0  become  rays  issuing  from  ?<•  =  0  at  twice  the  angle. 
(A  tigure  to  scale  should  l)e  supplied  by  the  reader.)  The  derivative 
U-'  =  2  z  vanishes  at  z  =  0  only.  Tlie  transformation  is  conformal  for 
all  points  except  z  =  0.  At  z  =  0  it  is  clear  that  the  angle  between 
two  curves  in  the  «-plane  is  douljled  on  passing  to  the  corresponding 
curves  in  the  ?/'-plane  ;  hence  at  z  —  0  the  transformation  is  not  con- 
formal.  Similar  results  would  l)e  obtained  from  vj  =  z""  except  that  the 
angle  between  rays  issuing  from  c  =  0  would  be  in  times  the  angle 
between  the  rays  at  z  =  0. 

A  point  in  the  neighborhood  of  whicli  a  function  ic  =  /(«)  is  ana- 
lytic but  has  a  vanishing  derivative  /''("-')  is  called  a  critical  point  of 
/(.t);  if  the  derivative  /'('-')  has  a  root  of  multiplicity  k  at  any  point, 
that  point  is  called  a  critical  point  of  order  k.  Let  z  =  s;^  be  a  critical 
point  of  order  /.-.    Expand  /'(-t)  as 

f(z)  =  a,{z  -  z^^  +  a,^,(z  -  z^f^^  +  a,,„X^  _  z^Y^^-  +  •  •  •  ; 
then  /(.)  =/(.,)  +  ^  (z  -  z^f^^  +  ^  (z  -  z;)^^-  +  •  •  • , 

or  y:  =  rr^  +  {z-z^>^-'E(z)     or     ,/•  _  ,r^  =  (^  _  ^j^+i£(-;),     (7) 

where  2s  is  a  function  that  does  not  vanish  at  z^.  The  })oint  z  =  ,-.•.  goes 
into  V  —  y^.  For  a  suthciently  small  region  al>out  z^^  the  transt'ornui- 
tion  (7)  is  sufficiently  represented  as 

"•-^'•o=^'(^--o)''S  ^'  =  ^(-o)- 

On  comparison  with  the  case  a-  =  z'",  it  appears  that  the  angle  between 
two  curves  meeting  at  z^^  will  V)e  multiplied  Ijy  k  -\-  1  on  jiassing  to  tin; 
corresponding  curves  meeting  at  ir^.  Hence  at  a  critical  point  of  t/ic 
kth  order  tlie  traiii^forinatlon  li<  not  conformad  hut  angles  are  riniltlplled 
hij  k  +  1  on  jjasslng  from  tlie  z-plane  to  tlie  icpAane. 

Consider  the  transformation  ir  =  ,-/-  ]n(jre  in  detail.  To  each  point  z 
corresponds  one  and  only  one  point  ir.  To  the  points  -;  in  tlie  first 
(juadrant  corres})ond  the  points  of  tlie  first  two  quadrants  in  the  v- 
lilane,  and  to  the  upper  half  of  the  .-.-plane  corres})onds  the  whole  ?/'-planc. 
In  like  manner  the  lower  half  of  the  -.--plane  will  be  mapped  u})on  the 
whole  ?/--plane.  Thus  in  finding  the  points  in  the  ?r-plane  Avhich  cor- 
resi>ond  to  all  the  yjoints  of  the  ;v-plane,  the  */--plane  is  covered  twice. 
This  double  counting  of  the  ?r-plane  may  be  obviated  by  a  simple  de- 
vice.    Instead  of  having  one  sheet  of  p;q)er  to  represent  the  */>plane, 


492 


THEORY   OF   FUNCTIONS 


let  two  sheets  be  superposed,  and  let  the  points  corresponding  to  the 
upper  half  of  the  2;-plane  be  considered  as  in  the  upj^er  sheet,  while 
those  corresponding  to  the  loAver  half  are  considered  as  in  the  lower 
sheet.  Now  consider  the  path  traced  npon  the  double  ?6"-plane  when  z 
traces  a  path  in  the  ;i-plane.    E\'ery  time  z  crosses  from  the  second  to 


n 


«-^otH} 


u 


w— surface 


w— surface 


z—X)lane 


the  third  (piadrant,  tv  passes  from  the  foui'th  quadrant  of  the  U])per 
sheet  into  the  first  of  the  lower.  AVhen  z  passes  from  the  fourth  to 
the  first  quadrants,  v  comes  from  the  fourth  quadrant  of  the  lower 
sheet  into  the  first  of  the;  iq)[)er. 

It  is  convenient  to  join  tlie  two  sheets  into  a  single  surface  so  that 
a  continuous  path  on  the  .'.'-plane  is  pictured  as  a  continuous  path  on 
the  ^r-surface.  This  may  be  done  (as  indicated  at  tlie  riglit  of  the 
middle  figui'c)  l)y  regarding  the  lower  half  of  the  u])pcr  sheet  as  con- 
nected to  the  u})[H',r  half  of  the  hjwer,  and  the  lower  half  of  the  lower 
as  connec'ted  to  the  upper  half  of  the  upper.  The  surface  therefore 
cuts  through  itself  along  the  })Ositive  axis  of  reals,  as  in  tlie  sketch  on 
the  left*;  the  line  is  called  \\\g  junrfioyi  line  of  the  siu'face.  The  point 
ir  =  0  whi('h  corresponds  to  the  ci'itical  point  z  =  0  is  called  the  hrdneh 
]>()lnt  of  the  sm'face.  Now  not  only  does  one  point  of  the  ,'v--])lane  go 
over  into  a  single  point  of  the  //--surface,  but  to  each  point  of  the  sur- 
face corresponds  a  single  jxdnt  ,-.■;  although  any  two  ])oints  of  the  //•- 
siu-face  which  are  super})os(Ml  have  the  same  value  of  v,  they  correspond 
to  different  values  of  ,-.-  except  in  the  case  of  tlu;  branch  point. 

184.  The  //'-surface,  which  has  been  obtained  as  a  mere  convenience 
in  mapping  the  .?-})]ane  on  the  //■-j)lane,  is  of  particular  value  in  study- 
ing the  inverse  function  ,-.'  =  V//-.  For  v //•  is  a  multiple  valued  func- 
tion and  to  each  value  of  //■  coi-resijond  two  values  of  ,'.':   but  if  //•  Ih* 


*  Practically  tliis  may  lie  accoin]ilislic<l  Un-  two  sheets  of  paper  l)y  pasting;  guniiiied 
striiis  to  the  sheets  wliich  ai-e  to  he  connected  across  the  cut. 


COMPLEX  VARIABLE 


493 


regarded  as  on  the  tr-surface  instead  of  merely  in  the  ^/--plane,  there  is 
only  one  value  of  ,~  corresponding  to  a  point  v:  upon  the  surface.  Thus 
the  function  ww  wliicJi  Is  double  valued  orer  the  w-phmc  beeoines  slnr/le 
valued  orer  the  w-surfare.  The  //--surface  is  called  the  lilemann  surfaee 
of  the  function  z  —  V/r.  The  construction  of  Riemann  surfaces  is  im- 
portant in  the  study  of  multi})le  valued  functions  because  the  surface 
keeps  the  different  values  apart,  so  that  to  each  point  of  the  surface 
corresponds  only  one  value  of  the  function.  Consider  some  surfaces. 
(The  student  should  make  a  paper  model  by  following  the  steps  as 
indicated.) 

Let  K'  —  2^  —  o2  and  plot  the  ?'>surface.  First  solve /'(2)  =  0  to  find  the  critical 
points  z  and  substitute  to  find  the  branch  points  c;.  Now  if  the  branch  points  be 
considered  as  removed  from  the  ?/.'-plane,  the  plane  is  no  longer  simply  connected. 
It  must  be  made  simply  connected  by  drawing  proper  lines  in  the  figure.  This  may 
be  accomplished  by  drawing  a  line  from  each  branch  point  to  infinity  or  by  con- 
necting the  successive  branch  points  to  each  other  and  connecting  the  last  one  to 
the  point  at  infinity.  These  lines  are  the  junction  lines.  In  this  particular  case  the 
critical  points  are  2:  =  +  1.  —  1  and  the  Viranch  points  are  (/•  =  —  2,  4-  2.  and  the 
junction  lines  may  be  taken  as  the  straight  lines  joining  v:  =  —  2  and  v  =  +  2  to 


I ,  II ,  III 

I  n  lu 

in  ni 

c'U 

(1 

-nC         "'-0 

L  ./ 

I'll' 

hi' 

b 

/ 

I'n'iii 

l',  11',  III' 

w- surface 

z- plane 


infinity  and  lying  along  the  axis  of  reals  as  in  the  figure.  N'ext  spread  the  requi- 
site number  of  sheets  over  the  !';-plane  and  cut  them  along  the  junction  lines.  As 
v:  =  2^  _  g  r  is  a  cubit'  in  2.  and  to  i^ach  value  of  v:.  except  the  branch  values,  there 
correspond  three  values  of  2.  three  sheets  are  needed.  Now  find  in  the  2-plane  the 
image  of  the  junction  lines.  The  junction  lines  are  represented  by  v  =  0  ;  but 
V  =  Sx-ij  —  y^  —  3  //.  and  hence  the  line  ?/  =  0  and  the  hyperbola  3.f-  —  y-  =  3  will 
be  the  images  desired.  The  z-plane  is  divided  into  six  pieces  which  will  l)e  seen  to 
correspond  to  the  six  half  sheets  over  the  i/>plane. 

Next  2  will  be  made  to  trace  (.>ut  the  images  of  the  junction  lines  and  to  turn 
about  the  crirical  points  so  that  ;/;  will  trace  out  the  junction  lines  and  turn  about 
the  brani;li  points  in  such  a  manner  that  the  connections  Vietween  the  different 
sheets  may  be  made.  It  will  be  c<invenient  to  regard  2  and  v:  as  persons  walking 
along  their  respective  paths  so  that  the  terms  "right ""  and  '"left""  have  a  meaning. 


4D4  THEORY  OF  FUXCTIOXS 

Let  z  start  at  z  =  0  and  move  forward  to  z  =  1 ;  tlien,  a.s/'(z)  is  negative,  vj  starts 
at  IV  =  0  and  moves  back  to  w  =  —  2.  Moreover  if  z  turns  to  the  right  as  at  P,  so 
must  w  turn  to  the  right  through  the  same  angle,  owing  to  the  conformal  property. 
Thus  it  appears  that  not  only  is  0^1  mapped  on  oa,  but  the  region  V  just  above  OA 
IS  mapped  on  tlie  region  Y  just  below  oa  ;  in  like  manner  OB  is  mapped  on  oh. 
x\s  ah  is  not  a  junction  line  and  the  sheets  have  not  been  cut  through  along  it,  the 
regions  1,  V  should  be  assumed  to  be  mapped  on  the  same  sheet,  say,  the  upper- 
most, I,  Y.  As  any  point  Q  in  the  whole  infinite  region  V  may  be  reached  from  0 
■without  crossing  any  image  of  ah.  it  is  clear  that  the  whole  infinite  region  1'  should 
be  considered  as  mapped  on  1' ;  and  similarly  1  on  I.  The  converse  is  also  evident, 
for  the  same  reast)n. 

If,  on  reaching  ^4,  tlie  point  z  turn.s  to  the  left  through  90°  and  moves  along  AC, 
then  w  will  make  a  turn  to  the  left  of  180°,  that  is,  will  keep  straight  along  ac ; 
a  turn  as  at  11  into  V  will  correspond  to  a  turn  as  at  r  into  I'.  This  checks  with 
the  statement  that  all  1'  is  mapped  on  all  I'.  Suppose  that  z  described  a  small 
circuit  about  -f-  1.  When  z  reaches  Z),  to  reaches  d  ;  when  z  reaches  E,  iv  reaches  c. 
But  when  v:  crossed  ac,  it  could  not  have  crossed  into  I,  and  when  it  readies  c  it 
cannot  be  in  I ;  for  the  points  of  I  are  already  accounted  for  as  corresponding  to 
points  in  1.  Hence  in  crossing  ac,  vj  must  drop  into  one  of  the  lower  sheets,  say 
the  middle,  II;  and  on  reaching  e  it  is  still  in  II.  It  is  thus  seen  that  II  corre- 
sponds to  2.  Let  z  continue  around  its  circuit;  then  IT  and  2'  correspond.  When 
z  crosses  AC  from  2'  and  moves  into  1,  the  point  lo  crosses  ac'  and  moves  from  II' 
up  into  I.  In  fact  tlie  upper  two  sheets  are  connected  along  ac  just  as  the  two 
sheets  of  the  surface  for  to  =  z-  were  connected  along  their  junction. 

In  like  manner  suppose  that  z  moves  from  0  to  —  1  and  takes  a  turn  alx)ut  B  so 
that  w  moves  from  0  to  2  and  takes  a  turn  about  h.  AVhen  z  crosses  BF  from  V  to  3, 
ic  crosses  //from  T  into  tlie  upper  half  of  some  .sheet,  and  this  must  be  III  for  the 
rea.son  that  I  and  II  are  already  mapped  on  1  and  2.  Hence  I'  and  III  are  con- 
nected, and  .so  are  I  and  III'.  This  leaves  II  which  has  been  cut  along  hf,  and  III 
cut  along  ac,  which  may  be  recoiniected  as  if  they  had  never  been  cut.  The  reason 
for  this  appears  forcibly  if  all  the  points  z  which  correspond  to  the  branch  points 
are  added  to  the  diagram.  When  ;/_•  =  2,  tlie  values  of  z  are  the  critical  value  —  1 
(double)  and  the  ordinary  value  z  =  2 ;  similarly,  \c  =  —  2  corresponds  to  z  =  —  2. 
Hence  if  z  describe  the  half  circuit  AE  .so  that  ic  gets  around  to  e  in  II.  then  if  z 
moves  out  to  z  =  2.  ic  will  move  out  to  lo  =  2,  pas.-^ing  liy  v:  =  0  in  the  sheet  II  as 
z  pa.'jses  through  z  —  ^  3  :  but  as  z  =  2  is  not  a  critical  point,  ic  =  2  in  II  cannot 
be  a  branch  point,  and  the  cut  in  II  may  be  reconnected. 

The  ('--surface  thus  coiistructed  for  ic  —f{z)  =  z^  —  3z  is  the  Riemann  .'surface 
for  the  inverse  function  z  =/-i(;c).  of  which  the  explicit  form  cannot  be  given 
without  solving  a  cubic.  To  each  point  of  the  surface  corres^ponds  one  value  of  z, 
and  to  the  three  superposed  values  of  vj  correspond  three  different  values  of  z  ex- 
cept at  the  branch  points  where  two  of  the  sheets  come  together  and  give  only 
one  value  of  z  while  the  third  sheet  gives  one  other.  The  I\iemann  surface  conM 
equally  ^vell  have  been  con.structed  by  joining  the  two  branch  points  and  tlun 
connecting  one  of  them  to  x.  The  image  of  r  =  0  would  not  have  been  cliangcil. 
The  connections  of  the  sheets  could  Ije  established  as  before,  but  would  Ik-  dif- 
ferent. If  the  junction  line  lie  —  2.  2.  +  x.  the  point  u:  =  2  has  two  junctions 
running  into  it.  and  the  connection.^  of  the  siieets  on  opposite  sides  of  the  point  are 
not  independent.   It  is  advisable  to  arrange  the  work  .<o  that  the  first  branch  point 


COMPLEX  VARIABLE  495 

which  is  encircled  sliall  have  only  one  junction  running  from  it.  This  may  be  done 
by  taking  a  very  large  circuit  in  z  so  that  w  will  describe  a  large  circuit  and  hence 
cut  only  one  junction  line,  namely,  from  2  to  cc,  or  by  taking  a  small  circuit  about 
z  =  1  so  that  w  will  take  a  small  turn  about  lo  =  —  2.  Let  the  latter  method  be 
choseii.  Let  z  start  from  z  =  0  at  0  and  move  to  z  =  1  at  A  ;  then  v;  starts  at  w  =  0 
and  moves  to  lo  =  —  2.  The  correspondence  between  V  and  I'  is  thus  established. 
Let  z  turn  about  A  ;  then  w  turns  about  w  =  —  2  at  a.  As  the  line  —  2  to  —  oo  or  ac 
is  not  now  a  junction  line,  v:  moves  from  I' 
into  the  upper  half  I,  and  the  region  across  \^v^^  V/^^' 
AC  from  1'  should  be  labeled  1  to  corre-  I  IN  "    \r 

spond.    Then  2',  2  and  II',  II  may  be  filled 
in.    The  connections  of  I-II'  and  II-I'  are 

indicated  and  III-III'  is  reconnected,  as  the  w— surface  z—j)lane 

branch  point  is  of  the  first  order  and  only  two 

sheets  are  involved.  Xow  let  z  move  from  z=:Otoz  =  —  1  and  take  a  turn  about 
B ;  then  lo  moves  from  w  =  0  to  w  =  2  and  takes  a  turn  about  6.  The  regioji  next 
1' is  marked  3  and  Y  is  connected  to  III.  Passing  from  .3  to  3'  for  z  is  equivalent 
to  passing  from  III  to  III'  for  lo  between  0  and  b  where  these  sheets  are  connected. 
From  3'  into  2  ffir  z  indicates  III'  to  II  across  the  junction  from  lo  =  2  to  oo.  This 
leaves  I  and  II'  to  be  connected  across  this  junction.  The  connections  are  com- 
plete. They  may  be  checked  by  allowing  z  to  describe  a  large  circuit  so  that  the 
regions  1,  1',  3,  3',  2,  2',  1  are  successively  traversed.  That  I,  I',  III,  III',  II,  II',  I 
is  the  corresponding  succession  of  sheets  is  clear  from  the  connections  between 
r/j  =  2  and  cc  and  the  fact  that  from  i«  =  —  2  to  —  oo  there  is  no  junction. 

Consider  the  function  w  =  z^  —  3z*  +  3z'^.  The  critical  points  are  z  =  0,  1,  1, 
—  1,-1  and  the  corresponding  branch  points  are  w  —  0,  1,  1,  1,  1.  Draw  the  junc- 
tion lines  from  v:  =  0  to  —  oo  and  from  wj  =  1  to  4-  oo  along  the  axis  of  reals.  To 
find  the  image  of  r  =  0  on  the  z-plane,  polar  coordinates  may  be  used. 

z  =  r(cos0  -I-  isin^),         w  =  w  +  n  =  r*'>e''"i"'  —  3r*e'**'  +  Sr'^e'^'t'K 

V  =  0  =  f^[)-*  sin  G(p  —  S  r-  sin  4  <;&  +  3  sin  2  0] 
=  r-  sin  2  4>[)-*{5  —  4 sin  2  0)  —  6  r-  cos  0  +  3]. 

The  equation  v  —  0  therefore  breaks  up  into  the  equation  sin  2  <p  =  0  and 

3cos2  0  ±  \/3sin2(;&       V3  sin  (60  ±2  0)  _  \/S 


3-4  An"  20  2    sin  (60  +  20)  sin  (60  -  2  0)       2  sin  (60  ±20) 


Hence  the  axes  0  =  0^  and  0  =  90"^  and  the  two  rectangular  liyperbolas  inclined  at 
angles  of  ±  15^  are  the  images  of  v  =  0.  The  z-plane  is  thus  divided  into  six  por- 
tions. The  function  v:  is  of  the  sixth  order  and  six  sheets  must  be  spread  over  the 
z/"-p]ane  and  cut  along  the  junction  lines. 

To  connect  up  the  sheets  it  is  merely  necessary  to  get  a  start.  The  line  u'  =  0 
to  w  =  1  is  not  a  junction  line  and  the  sheets  have  not  been  cut  through  along  it. 
But  when  z  is  small,  real,  and  increasing,  lo  is  also  small,  real,  and  increasing. 
Hence  to  OA  corresponds  oa  in  any  sheet  desired.  Moreover  the  region  above  OA 
will  correspond  to  the  upper  half  of  the  sheet  and  the  region  below  OA  to  the 
lower  half.  Let  the  sheet  be  chosen  as  III  and  place  the  numVjers  3  and  3'  so  as  to 
correspond  with  III  and  III'.    Fill  in  the  numbers  4  and  4'  aroniid  z  -  0.    When 


496 


THEORY  OF  FUNCTIONS 


z  turns  about  the  critical  point  2  =  0,  xo  turns  about  ?'j  3=  0,  but  as  angles  are  doubled 
it  nuist  go  around  twice  and  the  connections  III-IV,  IV-lIT  nuist  be  made.  Fill 
in  more  numbers  about  the  critical  point  z  =  1  of  the  second  order  where  angles  are 
tripled.  On  the  lo-sur- 
face  there  will  be  a 
triple  connection  III'-  I— YI 

II,  ir-I,  r-III.  In  ^„  =  , 
like  manner  the  criti-  \\  1  /// 
cal  point  2  =  —  1  may 
be  treated.  The  sur- 
face is  complete  except 
for  reconnecting  sheets 
I,II,V,  VI  along  !/;=0 
to  ;y  =  —  00  as  if  they 
had  never  been  cut.  Xf—auvfacQ  z— plane 


"TTT/^/ 


0 

iVvi' 

xv— surface 


EXERCISES 

1.  Plot  the  corresponding  lines  for:      (a)  v:  =  (1  +  2  i)z.         {j3)  v:  =  (1  —  ^^  i)z. 

2.  Solve  for  x  and  y  in  (1)  and  (2)  of  the  text  and  plot  the  corresponding  lines. 

3.  Plot  the  cf)rresponding  orthogonal  systems  of  curves  in  these  cases: 


(a)  I'J 


1 


(/3)  10  =  ]  +  Z-,         (7)   w  =  cos ; 


4.  Study  the  correspondence  between  z  and  70  near  the  critical  points: 

(a)   w  =  z^,         (/3)   !'j  =:  1  —  2'-,         (7)  !';  =  sin2. 

5.  Upon  the  i/>surface  for  c:  =  2-  plot  tlie  pnints  corresponding  to  2  =  1,1  +  ?, 
2  /,  —1-1-1  ^^t,  —  J,,  _  1  ^  ;j  _  1  /.  _  /.  I  —  I  i.  And  in_the  2-plane  plot  the 
points  corresponding  to  v:  =  V2  +  A  2 /,  (,  —  4,  —  J  —  ^  \'-'A,  1  —  (,  whether  in 
the  upper  ov  lower  sheet. 

6.  Construct  the  ir-surface  for  these  functions: 

(a)   w  =  2^,         (^)  w  =  z-  ■-'.  (7)   v:  =  1  +  z\         (5)  v-  =  (2  -  1)^. 

In  (li)  the  singular  point  2  =  0  should  be  joined  by  a  cut  t(.)  z  =  x. 

7.  Construct  the  IJieniaiin  surfaces  U>v  these  functions  : 

(a)    w  =  2^-2  2-,  (ii)   V  =  -  2^  +  4  2.  (7)   in  =  2z^ 


t)2-. 


(5)     K'  =  2  + 


>     1 

(e)    V  =  2-  +  -- 


(n  "• 


+  V?j : 


v:]2-  +  l 
185.  Integrals  and  their  inversion.    Consider  the  funetiou 

r"''hr  .  ,       . 


defined  li_v  au  iiiteL;-r;il,  ;nid  let  the  methods  of  the  tlieovy  of  fuiietions 
lu'  apjilied  to  tlie  study  of  the  function  tiiid  its  inverse.  If  w  deserihes 
a  ])ath  siuToundiiii,'  tlic  origin,  tlu'  integral  need  not  vanisli;    for  tlic 


z—x>lane  ic— plane 


COMPLEX  VARIABLE  497 

integiuucl  is  not  analytic  at  w  =  0.  Let  a  cut  be  drawn  from  w  =  0  to 
w  =  —  cc.  The  integral  is  then  a  single  valued  function  of  vj  provided 
the  path  of  integration  does  not  cross  the  cut.  jMoreover,  it  is  analytic 
except  at  w  =  0,  where  the  derivative,  which  is  the  integrand  1/ir, 
ceases  to  be  continuous.  Let  the  ?i'-plane  as  cut  be  mapped  on  the 
s-plane  by  allowing  w  to  trace  the  path  lahcdcfglill,  hy  computing  the 
value  of  ,v  sufficiently  to 
draw  the  image,  and  by 
applying  the  principles  of 
conformal  representation. 
"When  w  starts  from  w  =  1 
and  traces  1  a,  z  starts  from 
z  =  0  and  becomes  nega- 
tively very  large.  When  iv 
turns  to  the  left  to  trace  ah, 
z  will  turn  also  through  90° 
to  the  left.  As  the  integrand  along  ah  is  iW^,  z  must  be  changing  by  an 
amount  which  is  pure  imaginary  and  must  reach  li  when  ^v  reaches  b. 
When  IV  traces  he,  both  iv  and  die  are  negative  and  z  must  be  increasing 
by  real  positive  quantities,  that  is,  z  must  trace  BC.  AVheu  tr  moves  along 
cdefcj  the  same  reasoning  as  for  the  path  (d)  will  show  that  z  moves  along 
CDEFG.  The  remainder  of  the  path  may  be  com})lete(l  by  the  reader. 
It  is  now  clear  that  the  whole  ^r-plane  lying  between  tlie  infinitesimal 
and  infinite  circles  and  bounded  by  the  two  edges  of  the  cut  is  mapped 
on  a  strip  of  width  2  ttI  bounded  ui)on  the  right  and  left  by  two  infi- 
nitely distant  vertical  lines.  If  ir  ha,d  made  a  complete  turn  in  the  posi- 
tive direction  about  u-  =  0  and  returned  to  its  starting  ])oint,  z  Avould 
have  received  the  increment  2  iri  That  is  to  say,  tlie  values  of  z  which 
correspond  to  the  same  point  w  reached  by  a  direct  ])ath  and  by  a  path 
which  makes  k  turns  al)Out  w  =  0  will  differ  by  2  I-ttI.  Hence  when  w 
is  regarded  inversely  as  a  function  of  z,  the  function  will  be  periodic 
with  the  period  2  iri.  It  has  been  seen  from  the  correspondence  of 
cdef(/  to  CDEFG  that  tr  becomes  infinite  when  z  moves  off  indefinitely 
to  the  right  in  the  strip,  and  from  the  correspondence  of  BAIII  with 
ha  ill  that  u'  becomes  0  when  '.'  moves  off  to  the  left.  Hence  w  must  be 
a  rational  function  of  e".  As  y  neither  becomes  infinite  nor  vanishes 
for  any  finite  point  of  the  strip,  it  must  reduce  merely  to  Ce^'^  with  /.■ 
integral.  As  //•  has  no  smaller  period  than  2  ttI,  it  follows  that  k  =  1. 
To  determine  C,  compare  the  derivative  dw/dz  =  Ce^  at  ,-.■  =  0  with  its 
reciprocal  dz/dir  =  w~^  at  the  corresponding  point  iv  =  1;  then  C  =  1 
The  inverse  function  ln~^z  is  tlierefore  completely  determined  as  e'. 


498 


THEUHV   OF   FUNCTIONS 


In  like  manner  consider  tlie  integral 
dw 


Jr  «•      r, 


+  10- 


2  =/(«), 


4>{z)=f-Hz). 


B  AK  J 


Here  the  points  v;  =  ±  i  must  be  eliminated  from  the  !/>plane  and  the  plane  ren- 
dered simply  connected  by  the  proper  cuts,  say,  as  in  the  figure.  The  tracing  of 
the  figure  may  be  left  to  the  reader.  The 
chief  difficulty  may  be  to  show  that  the 
integrals  along  oa  and  be  are  so  nearly  equal 
that  C  lies  close  to  the  real  axis;  no  com- 
putation is  really  necessary  inasmuch  as  the 
integral  along  oc'  would  be  real  and  hence 
C  must  lie  on  the  axis.  The  image  of  the 
cut  ic-plane  is  a  .strip  of  width  tt.  Circuits 
around  either  +  i  f)r  —  i  add  tt  to  z,  and 
hence  w  as  a  function  of  z  has  the  period  tt. 
At  the  ends  of  the  strip,  lo  approaches  the 
finite  values  -|-  /  and  —  I.  The  fiuiction 
w  =  (p{z)  has  a  simple  zero  when  2  =  0  and 

has  no  other  zero  in  the  .strip.  At  the  two  points  z  =  ±  lir.  the  function  iv  becomes 
infinite,  but  only  one  of  these  points  .should  be  cniisidcre<l  as  in  the  strip.  As  the 
function  has  only  one  zero,  the  point  z  =  J,  ir  luust  be  a  pole  of  the  finst  order. 
The  function  is  therefore  completelj-  determined  except  for  a  constant  factor  which 
may  be  fixed  by  examining  the  derivative  of  the  function  at  the  origin.    Thus 


IV— plane 


1       1  e' 


+  1       i  e'^  +  c- 


=  tan  z. 


tan-ij/j. 


186.   As  a  tliird  example  consider  the  integral 


-[ 


■V  1  —  u- 


:f(u^,  ir  =  4>(rS)=f-\j:). 


(8) 


Here  the  inte^^'rand  is  double  valued  in  n-  and  eonsecjueiitly  there  is 
liable  to  be  confusion  of  the  tAvo  values  in  attempting  to  follow  a  path 
in  the  ?/'-plane.  Hence  a  two-leaved  surface  for  the  integrand  will  be 
constructed  and  the  path  of  integration  will  l)e  considered  to  be  on  the 
surface.  Then  to  each  ])oint  of  the  path  there  will  correspond  only  one 
value  of  the  integrand,  although  to  each  value  of  //•  there  correspond 
two  superimposed  points  in  tlu^  two  sheets  of  the  surface. 


As  the  radical  ^  1  —  ir~  vanishes  at  v}  =  ±\  and  takes  u\\  only  the  single  value  0 
instead  of  two  (Mjual  and  njipositc  values,  the  points  ir  =  -  1  are  Virancli  points  on 
the  surface  and  they  are  the  only  finite  branch  points.  S])read  two  sheets  over  the 
('.•-plane,  mark  the  branch  points  ic  =  —  1.  ami  draw  the  junction  line  between  them 
and  continue  it  (provisionally)  to  (/•  =  as.  At  /'•  =  —  ]  the  function  ^' 1  —  v:'-  may 
be  written  VI  -f-  v:  Kin-),  where  /•;  denotes  a  function  wliicli  does  not  vanish  at 
!'_•  =  — 1.  Hence  in  tlie  nei-liboihood  uf  ,/•  =  —  !  the  surfare  looks  like  that  for 
Vic  near  vi  =  0.    Tins  may  be  accomplished  by  making  the  connections  across  the 


COMPLEX   VARIABLE 


409 


junction  line.   At  tlie  point  w  =  +  1  the  surface  nuist  cut  througli  itself  in  a  similar 
manner.    This  will  be  so  provided  that  the  sheets  are  reconnected  across  Icxd  as  if 
never  cut ;  if  the  sheets  had  been  cross-connected  along  1  cc,  each  sheet  would  have 
been  separate,  though  crossed,  over  1,  and  the  branch  point  would 
have  disappeared.    It  is  noteworthy  that  if  w  describes  a  large  1  n 

circuit  including  both  branch  points,  the  values  of  Vl  —  w'^  are 
not  interchanged;  the  circuit  closes  in  each  sheet  without  pass- 
ing into  the  other.  This  could  be  expressed  by  saying  that  w  =  oo 
is  not  a  branch  point  of  the  function. 

Now  let  w  trace  out  various  paths  on  the  surface  in  the  attempt  to  map  the  sur- 
face on  the  z-plane  by  aid  of  the  integral  (8).  To  avoid  any  difficulties  in  the  way 
of  double  or  multiple  values  for  z  which  might  arise  if  w  turned  about  a  branch 
point  v:  =  ±  1,  let  the  surface  be  marked  in  each  sheet  over  the  axis  of  reals  from 

—  CO  to  +  1.  Let  each  of  the  four  half  planes  be  treated  separately.  Let  w  start 
at  w  =  0  in  the  upper  half  plane  of  the  ui)i>cr  sheet  and  let  the  value  of  Vl  —  v;- 
at  this  point  be  -f-  1 ;  the  values  of  VI  —  u:-  near  v:  =  0  in  IT  will  then  be  near 
-I-  1  and  will  be  sharply  distinguished  from  tiie  values  near  —  1  which  are  supposed 
to  correspond  to  points  in  1',  II.  As  w  traces  oa^  the  integral  z  increases  from  0  to 
a  definite  positive  number  a.  The  value  of  the  integral  from  a  to  b  is  infinitesimal. 
Inasmuch  as  lo  =  1  is  a  branch  point  where  two  sheets  connect,  it  is  natural  to 
assume  that  as  w  passes  1  and  leaves  it  on  the  right,  z  will  turn  through  half  a 
straight  angle.  In  other  words  the  integral  from  b  to  c  is  naturally  presumed  to  be 
a  large  pure  imaginary  affected 

with  a  positive  sign.    (This  fact        -5 — Q. Q. C_   D 

may  easily  be  checked  by  exam- 
ining the  change  in  Vl  —  u:'^ 
when  V.'  describes  a  small  circle 
about  w  =  1.  In  fact  if  tlie  E- 
function  -\^\  -j-  v;  be  discarded 
and  if  1  —  )/,'  be  written  as  re*', 
then  Vres*'  is  that  value  of  the 
radical  which  is  positive  when 
1  —  10  is  positive.  Now  when  w 
describes  the  small  semicircle, 
</)  changes  from  0^  to  —  180°  and  hence  the  value  of  the  radical  along  be  becomes 

—  i  Vr  and  the  integrand  is  a  positive  pure  imaginary.)  Hence  when  w  traces 
6c,  z  traces  BC.  At  c  there  is  a  right-angle  turn  to  the  left,  and  as  the  value  of 
the  integral  over  the  infinite  quadrant  cc'  is  J  tt,  the  point  z  will  move  back  through 
the  distance  \  tt.  That  the  point  ("  thus  reached  nuist  lie  on  the  pure  imaginary 
axis  is  seen  by  noting  that  the  integral  taken  directly  along  oc'  would  be  pure  imagi- 
nary. This  shows  that  a  =  |  tt  without  any  necessity  of  computing  the  integral 
over  the  interval  oa.    The  rest  of  the  map  of  I  may  be  filled  in  at  once  by  symmetry. 

To  map  the  rest  of  the  v.'-surface  is  now  relatively  simple.  For  V  let  v)  traee 
cc"d' ;  then  z  will  start  at  C  and  trace  CI)'  =  tt.  When  iv  comes  in  along  tlie  lower 
side  of  the  cut  d'e'  in  the  upper  sheet  I',  the  value  of  the  integrand  is  identical  with 
the  value  when  this  line  de  regarded  as  belonging  to  the  upper  half  plane  was  de- 
scribed, for  the  line  is  not  a  junction  line  of  the  surface.  The  trace  of  z  is  there- 
fore D'E'.  "When  v:  traces  f'o'  it  must  be  remembered  that  1'  joins  on  to  II  and 
hence  that  the  values  of  the  integrand  are  the  negative  of  those  along /'j.    This 


z— plane 


ic —surface 


500  THEORY   or  FUKCTIONS 

makes  z  describe  the  segment  F'O'  =  —  (r  =  —  |  tt.  The  turn  at  !!'¥'  checks  with 
tlie  straight  angle  at  the  branch  point  —  1.  It  is  further  notewortliy  tliat  when  w 
returns  to  o'  on  I',  z  does  not  return  to  0  but  takes  the  value  tt.  This  is  no  contra- 
diction; the  one-to-one  correspondence  which  is  being  established  by  the  integral 
is  between  points  on  the  w-surf  ace  and  points  in  a  certain  region  of  the  z-plane,  and 
as  there  are  two  points  on  the  surface  to  each  value  of  w,  there  will  be  two  points 
z  to  each  w.  Thus  far  the  sheet  I  has  been  mapped  on  the  z-plane.  To  map  II  let 
the  point  w  start  at  o'  and  drop  into  the  lower  sheet  and  then  trace  in  this  sheet 
the  path  which  lies  directly  under  the  path  it  has  traced  in  I.  The  integrand  now 
takes  on  values  which  are  the  negatives  of  those  it  had  previously,  and  the  image 
on  the  2-plane  is  readily  sketched  in.  The  figure  is  self-explanatory.  Thus  the 
complete  surface  is  mapped  on  a  strip  of  width  2  tt. 

To  treat  the  different  values  which  z  may  have  for  the  same  value  of  to,  and  in 
particular  to  determine  the  periods  of  w  as  the  inverse  function  of  z,  it  is  necessary 
to  study  the  value  of  the  integral  along  different  sorts  of  paths  on  the  surface. 
Paths  on  the  surface  may  be  divided  into  two  classes,  closed  paths  and  those  not 
closed.  A  closed  path  is  one  which  returns  to  the  same  point  on  the  surface  from 
which  it  started  ;  it  is  not  sufficient  that  it  return  to  the  same  value  of  w.  Of  paths 
which  are  not  closed  on  the  surface,  those  which  close  in  w,  that  is,  which  i-eturn 
to  a  point  superimposed  upon  the  starting  point  but  in  a  different  sheet,  are  the 
most  important.  These  paths,  on  the  particular  surface  here  studied,  may  be  fur- 
ther classified.  A  path  which  closes  on  the  surface  may  either  include  neither 
branch  point,  or  may  include  both  branch  points  or  may  wind  twice  around  one 
of  tlie  points.  A  path  which  closes  in  w  but  not  on  the  surface  may  wind  once 
about  one  of  the  branch  points.    Each  of  these  types  will  be  discussed. 

If  a  closed  path  contains  neither  branch  jwiut,  there  is  no  danger  of  confu.sing 
the  two  values  of  the  function,  the  projection  of  the  path  on  the  w-plane  gives  a 
region  over  which  the  integrand  may  be  considered  <as  single  valued  and  analytic, 
and  hence  the  value  of  the  circuit  integral  is  0.  If  the  ^ivdh  surrounds  both  branch 
points,  there  is  again  no  danger  of  confusing  the  values  of  the  function,  but  the 
projection  of  the  path  on  the  lo-plane  gives  a  region  at  two  points  of  which,  namely, 
the  branch  jjoints,  the  integrand  ceases  to  be  analytic.  The  inference  is  that  the 
value  of  the  integral  may  not  be  zero  and  in  fact  will  not  lie  zero  unless  the  in- 
tegral around  a  circuit  shrunk  close  up  to  the  branch  points  or  exjjanded  out  to 
infinity  is  zero.  The  integral  around  cv'dc/'c  is  here  equal  to  2  7r;  the  value  of  the 
integral  around  any  path  which  incloses  both  branch 
points  once  and  only  once  is  therefore  2  tt  or  —  2  tt  ac- 
cording as  the  path  lies  in  the  upper  or  lower  .sheet  ;  if 
the  path  surrounded  the  points  k  times,  the  value  of 
the  integral  would  be  Ikiz.  It  thus  ajtpears  that  w  re- 
garded as  a  function  of  z  has  a  period  2  tt.  If  a  path 
closes  in  v)  but  not  on  the  surface,  let  the  point  where  it 

crosses  the  junction  line  be  held  fast  (figure)  while  the  path  is  shrunk  down  to 
whiui'b'w.  The  value  of  the  integral  will  not  change  during  tliis  shrinking  of  the 
])atli,  for  thy  new  and  (jld  paths  may  together  be  regarded  as  closed  and  of  the 
first  case  considered.  Almig  tin'  paths  vha  and  n'h'w  the  integrand  has  opposite 
signs,  but  so  has  ihv;  ai-ound  the  small  circuit  the  value  of  the  integral  is  infini- 
tesimal. Hence  the  value  of  the  integral  around  the  path  which  closes  in  w  is  2  I 
or  —  2  I  if  I  is  the  value  from  the  point  a  where  the  path  crosses  the  junction  line 


COMPLEX  VARIABLE 


501 


to  the  point  iv.  The  same  conclusion  would  follow  if  the  path  were  considered  to 
shrink  down  around  the  other  branch  point.  Thus  far  the  possibilities  for  z  corre- 
sponding to  any  given  w  are  z  +  2k7r  and  2  7mr  —  z.  Suppose  finally  that  a  path 
turns  twice  around  one  of  the  branch  points  and  closes  on  the  surface.  By  shrink- 
ing the  path,  a  new  equivalent  path  is  formed  along  which  the  integral  cancels  out 
term  for  term  except  for  the  small  double  circuit  around  ±  1  along  which  tlie 
value  of  the  integral  is  infinitesimal.  Hence  the  values  z  +  2kTr  and  2m7r  —  z  are 
the  only  values  z  can  have  for  any  given  value  of  w  if  z  be  a  particular  possible 
value.  This  makes  two  and  only  two  values  of  z  in  each  strip  for  each  value  of  iv, 
and  the  function  is  of  the  second  order. 

It  thus  appears  that  iv,  as  a  function  of  z,  has  the  period  2  7r,  is  single  valued, 
becomes  infinite  at  both  ends  of  the  strip,  has  no  singularities  within  the  strip,  and 
has  two  simple  zeros  at  z  =  0  and  z  =  tt.  Hence  w  is  a  I'ational  function  of  e'«  with 
the  numerator  e~"—  1  and  the  denoniinatur  c-''-  -\-  1.    In  fact 


w=C 


1  e'-  —  c 


e"=  +  e- 


i  t'~  -\-  Q- 


The  function,  as  in  the  previous  cases,  has  been  wholly  determined  by  the  general 
methods  of  the  theory  of  functions  without  even  computing  a. 
One  more  function  will  be  studied  in  brief.    Let 


=  X" 


dw 


{a  —  iv)  Viv 


«>0, 


=  f{w),        w  =  4>{z)=f-\z). 


Here  the  Kiemami  surface  has  a  branch  point  at  jy  =  0  and  in  addition  there  is  the 
singular  point  w  =  a  of  the  integrand  which  must  be  cut  out  of  both  sheets.    Let 
the  surface  be  drawn  with  a  junction  line  from  w  =  0  to  iu  =  —  co  and  with  a  cut 
in  each  sheet  from  w  =  a  to  w  =  cc.   The 
map  on  the  z-plane  now  becomes  as  indi- 
cated in  the  figure.    The  different  values 
of  z  for  the  same  value  of  lo  are  readily 
seen   to   arise  when    w    turns  about   the 
jioint   10  =  a  in  either  sheet  or   when  a 
path  closes  in  w  but  not  on  the  surface. 
These  values  of  z  are  z  +  2k7ri/Va  and 
2imri/\a  —  z.    Hence  iv  as  a  function  of 
z  has  the  period  2  7ria~2,   has  a  zero  at 

z  =  0  and  a  pole  at  z  =  iri/Va,  and  approaches  the  finite  value  w  =  a  at  both  ends 
of  the  strip.  It  must  be  noted,  however,  that  the  zero  and  pole  are  botli  neces- 
sarily double,  for  to  any  ordinary  value  of  iv  correspond  two  values  of  z  in  the 
strip.    The  function  is  therefore  again  of  the  second  order,  and  indeed 


z— plane 


w— surface 


(e^ 


Vcl. 


1)-^ 


(e^ 


Va 


+  1) 


=  a  tanh-  ~  z  Va, 
2 


tanh-i 


V^ 


The  success  of  this  method  of  determining  the  function  z  =f{iv)  defined  by  an 
integral,  or  the  inverse  w  =f-'^{z)  =  ^{z),  has  been  dependent  tirst  upon  the  ease 
with  which  the  integral  may  be  used  to  map  the  jo-plane  or  ?c-suvface  upon  the 
z-plane,  and  second  upon  the  simplicity  of  the  map,  which  was  such  as  to  indi- 
cate that  the  inverse  function  was  a  single  valued  pericjdic  function.    It  should  be 


502  THEORY  OF  FUNCTIONS 

realized  tliat  if  an  attempt  were  made  to  apply  the  methods  to  integrands  which 
appear  equally  simple,  say  to 

2  =   I    A  (/'-  —  w-dH\         z=   I   {a  —  ic)  dv:/\'v:, 

the  method  would  lead  only  with  great  difficulty,  if  at  ail,  to  the  relation  between 
2  and  w  ;  for  the  functional  relation  between  z  and  w  is  indeed  not  simple.  There 
is,  however,  one  class  of  integrals  of  great  importance,  namely, 

dw 


f 


A  (w  —  cx^){iv  —  (Xr,)  ■  ■  •  (w  —  a„) 
for  which  this  treatment  is  suggestive  and  useful. 

EXERCISES 

1.  Discuss  by  the  method  of  the  theory  of  functions  these  integrals  and  inverses  ; 
f-  «■  div  ,  ,     r'"  2  dio  ,  ,     r"'     dw 

J I      '2  10  «^ii      1  —  w  Jo      1  —  R- 

r"'       dw  ,  ,     r"'       dw  ,  ,     r"'         dw 

(5)       I  •  (0      /        -— :=.  (n      /         J-^ 


+ 
The  I'esuils  may  be  checked  in  each  case  by  actual  integration. 
dw  ,     /"'•'      dw 


r"'  dw  ,,,      r'"  (Zi'j  ,   ,     r"'  dw 


X"'              dii}                          r "-'      '(('' 
—     -    '     and     I     ' (i;  182.  and  Ex.  10.  p.  489). 

-      A    W{\-  W){\  +    W)  ^'"      ■\    1-  i/J* 


CHAPTER   XIX 


ELLIPTIC  FUNCTIONS  AND  INTEGRALS 


187.  Legendre's  integral  I  and  its  inversion.    Consider 


£ 


(/,(■ 


V(l  -  ir-)  (1  -  Jric^) 


0  <k  <  1. 


(I) 


The  Kiemann  surface  for  the  integrand*  has  branch  points  at  ir  =  -j-  1 
and  ±  1/k  and  is  of  two  siieets.  Junction  lines  may  l)e  drawn  between 
+  1,  +  1/k  and  —  1,  —  1//.-.  For  very  hirge  values  of  w,  the  radical 
Vi^l  —  ir-)  (1  —  Irir'-)  is  approximately  ±  h-ir-  and  hence  there  is  no 
danger  of  confusing  the  values  of  the  function.  Across  the  junction 
lines  the  surface  may  be  connected  as  indicated,  so  that  in  the  neigh- 
borhood of  tr  =  +  1  and  ((•  =  ±  1/k  it  looks  like  the  surface  for  'wir. 
Let  +  1  be  the  value  of  the  integrand  at  tc  =  0  in  the  upper  sheet. 
Further  let 


K  = 


(hi- 


V(i  —  ((•-)  (1  —  k'-( 


!h"  = 


') 


f: 


dt 


V(l  —  ir-)  (1  —  Irir-) 


(1) 


Let  the  changes  of  the  integral  be  followed  so  as  t(^  map  the  surface 
on  the  ,v-})lane.  As  w  moves  from  t>  to  <■/,  the  integral  (I)  increases 
by  A',  and  .-  moves 


from  O  to  .1 .  As  ^r 
continues  straight 
on,.-;  nuikes  a  right- 
angle  turn  and  in- 
creases by  pure 
imaginary  incre- 
ments to  the  total 
amount  IK'  Avhen 
V  reaches  A.  As  ic 
continues  there  is 


E     D 


F 


C  B 
1 

O  A 

0' 

1' 

2 

z— plane 


IV— surface 


another  right-angle  turn  in  z,  the   integrand  again  becomes  real,  and 
,•;  moves  down  to  C.    (That  z  reaches  C  follows  from  the  facts  that  the 

*  Tlie  reader  unfjuniliar  with  Rieniaiiii  surfaces  ("§  1S4)  may  proceed  at  once  to  identify 
(1)  aud  ("2)  by  Ex.  9,  p.  47,")  and  may  take  (1)  and  other  nece.ssary  statements  for  granted. 

503 


504  THEORY   OF  FUNCTIONS 

integral  along  an  infinite  (j^uadrant  is  infinitesimal  and  that  the  direct 
integral  from  0  to  ix  would  be  pure  imaginary  like  <hc.)  If  ic  is  allowed 
to  continue,  it  is  clear  that  the  map  of  I  will  be  a  rectangle  2  A'  by  A'' 
on  the  -.'-plane.  The  image  of  all  four  half  planes  of  the  sui'facc  is  as 
indicated.  The  conclusion  is  reasonably  apparent  that  w  as  the  inverse 
function  of  z  is  doubly  periodic  with  periods  4  A  and  2  iK\ 

The  periodicity  may  be  examined  more  carefully  by  considering  different  possi- 
bilities for  paths  upon  the  surface.  A  path  surrounding  the  pairs  of  branch  points 
1  and  A;-^  or  —  1  and  —  k-'^  will  close  on  the  surface,  but  as  the  integrand  has  oppo- 
site sigTis  on  opposite  sides  of  the  junction  lines,  the  value  of  the  integral  is  2  iK'. 
A  path  surrounding  —  1,  +  1  will  also  close ;  the  small  circuit  integrals  about  —  1 
or  ■{■  1  vanish  and  the  integral  along  the  whole  path,  in  view  of  the  opposite  values 
of  the  integrand  along /<(  in  I  and  II,  is  twice  the  Integral  f  rom  /  to  a  or  is  4  A. 
Any  path  which  closes  on  tlie  surface  may  be  resolved  into  certain  multiples  of 
these  paths.  In  addition  to  paths  which  close  on  the  surface,  paths  which  close  in 
w  may  be  considered.  Such  paths  may  be  resolved  into  those  already  mentioned 
and  paths  running  directly  between  0  and  w  in  the  two  sheets.  All  possible  values 
of  z  for  any  w  are  therefore  4  niK  -\-  2  niK'  ±  z.  The  function  w  (z)  has  the  periods 
4  K  and  2  iK\  is  an  odd  function  of  z  as  iv{—  z)  =  lo  (z),  and  is  of  the  second  order. 
The  details  of  the  discussion  of  various  paths  is  left  to  the  reader. 

Let  w  =f(z).  The  function  f('S)  vanishes,  as  may  be  seen  by  the 
map,  at  the  two  points  -.■  =  0,  2  A  of  the  rectangle  of  periods,  and  at 
no  other  points.  These  zeros  of  a-  are  simple,  as  /'('■•')  does  not  vanish. 
The  function  is  therefore  of  the  second  order.  There  are  poles  at 
z  =  /A',  2  A'  -f  IK',  Avhieh  must  be  simple  poles.  Finally /'(7v)  =  1.  The 
])Osition  of  the  zeros  and  poles  determines  the  function  except  for  a  con- 
stant multi})lier,  and  that  will  be  fixed  by  f(K)  =  1 ;  the  function  is 
wholly  determined.  The  function  /"(■-)  ^^^^Y  ^^^^^'  '-"^  identified  with  sn  z 
of  §  177  and  in  particular  with  the  special  case  for  which  K  and  A'  are 
so  related  that  the  niulti})lier  ;/  =  1. 

Q(K)  II (z) 
w  =  t(z)  =  — ' =  sn  ,-;,  ,•:;  =  v.  (J) 

For  the  quotient  of  the  theta  fiuictions  has  simple  zeros  at  0,  2  7\', 
where  the  nunu'rator  vanishes,  and  simple  poles  at  iK' .  2  A'  +  'A',  where 
the  denominator  vanislies;  tlii'  (juotieiit  is  1  at  ,-.'  =  A;  and  the  deriva- 
tive of  sn  z  at  ,-.:  =  0  is  y  en  0  dn  0  —  <j  =  1,  whereas /''(Oj  =  1  is  also  1. 
The  imposition  of  the  cDiidirion  y  =  1  was  seen  to  im])Ose  a  relation 
between  A',  7\''.  /■,  /.',  y  by  vii-tue  of  wliicli  oidy  one  of  the  five  remained 
inde]iendent.  The  definition  of  A' and  A''  as  definite  integrals  also  makes 
them  functions  K(^l:)  and  A'(7.'^  of  /.-.    But 


1 


ELLIPTIC  FUNCTIONS  505 


V(l  -  ,r^)  (1  -  k'^c^) 

i      V(l  -  .rf)  (1  - //Vf)  ^    ^ 

if  w  =  (1  —  J:''-w\y-  and  /.•"-  +  ^''^  =  1-  Hence  it  appears  that  K  may  be 
computed  from  A-'  as  A''  from  U.  This  is  very  useful  in  practice  when 
}r  is  near  1  and  A-'-  near  0.    Thus  let 

-r-^;         ,      11-  Va       2/1-Va\'  .         ,        ,         o 

«    "^  =  7  =  o  : ^T  +  .>3    : /T    +  •  •  • '       ^"^^  y  1"J.^  y'  =  tt', 


-'  1  +  V/.-     -''  \i  +  V/.v  ^^^ 

^jf^'  =0:(O,  7')  =  1  +  2y'  +  2v'^  +  ..  .,  A-  =  -  ^^'log/; 

and  compare  with  (37)  of  \^.  472.  Now  either  /..•  or  /.■'  is  greater  than  0.7, 
and  hence  either  y  or  7'  may  be  obtained  to  five  places  with  only  one 
term  in  its  expansion  and  Avith  a  relative  error  of  only  about  0.01  per 
cent.  ^Moreover  either  </  or  //'  will  be  less  than  1/20  and  hence  a  single 
term  1  +  2  y  or  1  +  2  y'  gives  A'  or  A'  to  four  places. 

188.  As  in  the  relation  between  the  liiemann  surface  and  the  ,v-plane 
the  whole  real  axis  of  ,-;  corresponds  periodically  to  the  part  of  the  real 
axis  of  v  between  —  1  and  +  1,  the  function  sn  .-r,  for  real  x,  is  real. 
The  graph  of  ^  =  sn  a-  has  roots  at  .r  =  2  ///A',  maxima  or  minima  alter- 
nately at  (2  tiL  +  Ij  A",  inflections  inclined  at  the  angle  45°  at  the  roots, 
and  in  general  looks  like  y  =  sin  (ttx /2  K).  Examined  more  closely, 
sn  i  A'  =  (1  +  ]:' )~  -  >  2~  -  =  sin  \  ir ;  it  is  seen  that  the  curve  sn  x  has 
ordinates  numerically  greater  than  sin  {■nx]'!  K).    As 

en  .'/•  =  V 1  —  sn-  X,  dn  ./■  =  Vl  —  Ir  su"  ./•,  (5) 

the  curves  //  =  en  x.  //  =  dn  ./■,  may  readily  be  sketched  in.  It  may  be 
noted  that  as  sn  (./■  +  A)  ^  en  x,  the  curves  for  sn  x  and  en  x  cannot 
be  superposed  as  in  tlie  case  of  the  trigoncjmetric  functions. 

The  segment  0,  iK'  of  the  })ure  imaginary  axis  for  z  corres})Onds  to 
the  whole  upper  half  of  the  pure  imaginary  axis  for  ir.  Hence  sn  ix 
with  ./•  real  is  pure  imaginary  and  —  l  sn  /,/•  is  real  and  })()sitive  for 
0  s  ,,■  <  A' '  and  Ijecomes  infinite  for  ./■  =  A''.  Hence  —  l  sn  /,/•  looks  in 
general  like  tan  (ttx  j2  1\).  V>\  (5)  it  is  seen  that  the  curves  for  y  =  en  Ix, 
y  =  dn  /./•  look  much  like  sec  ('Trx/2  K')  and  that  en  Ix  lies  above  dn  ix. 
These  functions  are  real  for  pure  imaginary  values. 

It  was  seen  that  when  /.-  and  /.•'  interchanged.  A'  and  A'  also  inter- 
changed. It  is  therefore  natural  to  look  for  a  relation  betAveen  the  ellip- 
tic functions  sn  (.v,  A),  en  (.v,  /.■),  dn  (-.-,  /.•)  formed  Avith  the  modulus  k 


506  THEORY   OF  FUNCTIONS 

and  the  functions  sn  {z,  k'),  cu  (z,  k'),  dn  (z,  k')  formed  with  the  com- 
plementary modulus  k'    It  will  be  shown  that 

.  sn  (z,  k')  .sn(lz,k') 

sn  (iz,  k)  =  I  — y-ri  '         sn  (z,  A')  =  -  «  — )-. — -f , 
^         -^         en  (z,  k )  ^  en  (<,?,  k) 

1  1 

en  {iz,  k)  =  — — — 77-  J  en  (z,  k) 


en  (z,  k')  ^  '    ^       en  (Iz,  k') 

Consider  sn  (Iz,  k).  This  function  is  periodic  with  the  periods  4 /v  and 
2  IK'  if  Iz  be  the  variable,  and  hence  with  periods  4  iK  and  2  A''  if  z  be 
the  variable.  With  z  as  variable  it  has  zeros  at  0,  2  /A',  and  poles  at 
A'',  2  iK  +  A''.  These  are  precisely  the  positions  of  the  zeros  and  poles 
of  the  quotient  H(z,  q')/H^{z,  q'),  where  the  theta  functions  are  con- 
structed with  q'  instead  of  q.  As  this  quotient  and  sn  (iz,  k)  are  of  the 
second  order  and  have  the  same  periods, 

^    '^"      Z//-^,'/)       "^^  en  (.,;.•') 
The  constant  C^  may  be  determined  as  C\  =  i  by  comparing  the  deriva- 
tives of  the  two  sides  at  z  =  0.  The  other  five  relations  may  be  proved 
in  the  same  way  or  by  transformation. 

The  theta  series  converge  with  extreme  rapidity  if  q  is  tolerably 
small,  but  if  q  is  somewhat  larger,  they  converge  rather  poorly.    The 
relations  just  obtained  allow  the  series  with  q  to  be  replaced  by  series 
with  q'  and  one  of  these  quantities  is  surely  less  than  1/20. 
In  fact  if  V  =  7rx/2  K  and  v'  =  7r.r/2  A",  then 

_  Vy        2  sin  v  —  2  y-  sin  3  v  -f-  2  q'''  sin  5  v  —  •  •  • 
sn  (X,   .  j  —  ^-  ^__  .j^^  ^.^^  2  V  +  2  7'  cos  4  V  —  2  y''  cos  G  v  H 

_    1     sinh  v'  —  '/'"  sinh  3  v'  +  7"'  sinh  0  v'  —  •  ■  • 
V/.-  *-'Osh  v'  +  q''  cosh  3  v'  +  q"'  cosh  o  v'  +  •  •  • 

The  second  series  has  the  disadvantage  that  the  hyperbolic  functions 
increase  rapidly,  and  hence  if  the  convergence  is  to  Ijc  as  good  as  for 
the  first  scries,  the  value  of  y'  must  be  considerably  less  than  that  of 
y,  that  is,  A'  must  be  consideral)ly  less  than  A'.  This  can  readily  be 
arranged  for  work  to  four  or  five  places.    For 

y'«  =  e~  '^~^',  cosh  .")  v'  =  1  (p'-'''  +  e'^') ,         0  ^  ./•  ^  A", 

where  owing  to  the  periodicity  of  the  functions  it  is  never  necessary 
to  take  ./•  >  A''.   The  term  in  q"''  is  therefore  less  than  }j  y'"-.    If  tlie  term 


ELLIPTIC   FUXCTIOXS  507 

in  '/'*'  is  to  be  equally  negligible  Avitli  that  in  (f, 

2  '/"  =  i  7'-     with     log  q  log  q'  =  tr-, 
from  Avhich  q'  is  deterniinecl  as  about  q'  =  .02  and  q  as  about  q  =  .08; 
the  neglected  term  is  about  0.0000005  and  is  barely  enough  to  effect 
six-place  work  except  through  the  multiplication  of  errors.    The  value 
of  ic  corresponding  to  this  critical  value  of  q  is  about  A-  =  0.85. 
Another  form  of  the  integral  under  consideration  is 
f  '^  (W  _    r"  da- 


sin  (f)  =  >/  =  sn  .'•,         (f>  =  am  ,/■,         cos  (f>  =  v  1  —  sn-  x  =  en  x, 


\cf>  =  Vl  -  /.•■-//-  =  Vl  -  /.-  sin-"^  =  dn  ,/•,         /.■'-  =  1  -  Ir, 
X  =  sn-\,y,  A-)  =  cn-i(Vl  -  if,  /.•)  =  dn-Y^-^l  -  /r'A  /.')• 
The  angle  <f>  is  called  the  amplltudu  of  ,/• :  the  functions  sn  .r,  en  a', 
dn  X  are  the  slne-ampIltude,  coslne-amjilitudi',  djdt<i-(i inplltude  of  x.   The 
half  periods  are  then 


K=\  .       '"  =F(Uk 


'  ~  J,       Vl  -  fr  sin^ 

Jo        Vl-A-'-^sin-^^  \-^  / 


(8) 


and  are  knoAvn  as  the  complete  e//ipf!c  inffr/ra/s  of  the  first  hind. 

189.  The  elliptic  functions  and  integrals  often  arise  in  problems 
that  call  for  a  numerical  answer.  Here  /.-  is  given  and  the  conq^lete 
integral  K  or  the  value  of  the  elliptic  functions  or  of  the  elliptic  inte- 
gral F(</>.  /.■)  are  desired  for  some  assigned  argument.  The  values  of 
A'  and  Fi^.  /■•)  in  terms  of  sin~V.-  are  found  in  tal)les  (B.  0.  Peirce, 
pp.  117-119),  and  may  be  obtained  therefrom.  The  tables  may  be 
used  by  inversion  to  find  the  values  of  the  function  sn  ./■,  en  .v.  dn  ,/- 
when  X  is  given  ;  for  sn  x  =  sn  F[<^.  /,■)  =  sin  <^.  and  if  ./•  = /•"  is  given, 
(^  may  be  found  in  the  table,  and  tlicn  sn  ./•  =  sin  <^.  It  is,  however, 
fa>y  to  com})ute  tlu^  desired  values  directly,  owing  to  tlie  extrenie 
rajiiditv  of  the  conv(*rgence  of  the  series.    Thus 

VTk  (2  A7j'  1  +  VP     /—       1 

—  ^  0  (T)).     ^ =  0(0),     — -^=~  VA'  =  -(0/0;  +  0(0)), 


^^^'  =  J^(l+2,*+...)  =  .J-flog,'  (9) 

^V-2logy;.^       o     „^..  , 

1  +  Va 


508  TJIEOllY   OF  FUNCTION'S 

The  elliptic  functions  are  computed  from  (6)  or  analogous  series. 
To  compute  the  value  of  the  elliptic  integral  F  ((f>,  /.■),  note  that  if 

(In  X       1  +  2  q  cos  2  V  +  2  7*  cos  4  v  +  •  •  • 

cot  A  =  -7=-  =  z.-^^ „    ;  o  % — r"-r — '        (lo) 

VA;'        1  —  2  y  cos  2v  +  2tf  cos  4  v  +  •  •  •  ^     ^ 

/I  \      cot  X  —  1       ^     cos  2  V  +  v^  cos  G  V  ^ 

tan    -  TT  —  A    =  — — T  =  2  y  — ;—-, ; 

\4  /      cot  A  +  1  ^      1  +  2f/  cos  4  V  H 

2 '/  cos  2  V 

and    tan  (i  tt  -  A)  =  2  y  cos  2  v  or  tan  (j  tt  -  A)  =  :; fr-r —  (10') 

^■*  ^  '  ^*  ^      1  +  2  y^  cos  4  V  ^      ^ 

are  two  approximate  equations  from  which  cos  2  v  may  be  obtained ; 
the  first  neglects  y*  and  is  generally  sufficient,  but  the  second  neglects 
only  yl    If  k'^  is  near  1,  the  proper  approximations  are 

1     dn  (.r,  Z-)  _  dn  (^.r,  /.■')       1  +  2  y' cosh  2  v' +  •  ■  ■ 
""  -v^  en  (.r,  A:)  ~        VZ^  1  -  ^  y'  cosh  2  v'  +  •  •  ■ '    ^^^^ 

tana7r-A)  =  2y'cosh2v'  or  tan  Q  tt  -  A)  =  ^  _^  j^'^;"^||  J^"j  ^,  ■  (11') 

Here  y'^  cosh  8  v'  <  y'^  is  neglected  in  the  second,  but  y'*  cosh  4  v'  <  y'" 
in  the  first,  which  is  not  always  sufficient  for  four-place  work.  Of  course 
if  <^  with  sn  .r  =  sin  (/>  or  if  //  =  sn  x  is  given,  dn  x  —  Vl  —  /.•-  sn'-^  x  and 
en  x  =  V 1  —  sn'-cc  are  I'cadily  computed. 


As  an  exaiiiT 


iple  take  , --^=  and  fnid  K,  sn  I  A',  F(J  tt.  i).    As  k'-  =  J- 

and   V/(;'>0.0,  the  lirst   term  of  (87),  p.  472,  gives  q  accurately  to  five  places. 
Compute  in  the  form :   (Lg  =  h)gjp) 

Lg  A;'2  =  9.87.50(5  Lg  (l  -  VIT')  =  8.84136  Lg  2  tt  =  0. 7982 

Lg  VP  =  9.9087(5  Lg  (l  +  v/p)  =  0.28569  2  Lg  (l  +  W?)  =  0.571 4 

VF  =  9.93060  Lg  2  y  =:  8.55-567  Lg  K  =  0.22(58 

1  -  VP  =  0.06940  2  7  =  0.03595  K  =  1 .686 

1  +  Va?  =  1.93060  q  =  0.01797  Check  with  table. 

2  „      ^^  ^/y  sin  Itt  —  y^  sin  tt  +  •  •  •       „  ^/y  i  Vs 
sn  -  7v  =  2  — £ ^ =  2  — - • 

3  VA;     1  —  2  y  cos  ;:|  tt  +  •  ■  •  -y^  1  +  '/ 

2    ,       VO  \/y  1  Lg  6  =  0.38908         Lg  sn  2  K  =  9.9450 

^"  3   ^  ^  1.01797  1  Lg  q  =  9.5(53(56  .sn  |  A'  =  0.8810. 

-  Lg  1.018  =  9.99226 


Af  =  dn  ,r  =  v  1  —  |  sin-  J  tt  =  v^l  —  J  sin  J  tt  V  1  +  I  sin  J  tt. 


ELLIPTIC  FUNCTIONS 


509 


I  sin  i7r  =  0.19134 

1  -  i  sin  1  IT  =  0.80866 

1+  isinl7r=  1.19134 

I  Lg  (1  -  I  sin  1  tt)  =  9.95388 

I  Lg  (1  +  i  sin  1  tt)  =  0.03802 

-  Lg  vF  =  0.03124 

Lg  cot  X  =  0.02314 


X  =  43°  28'  28'' 
i  TT  -  X  =  1°  31'  32" 
Lg  tan  =  8.42540 
Lg2(/  =  8.55567 
Lgcos2;'  =  9.86973 
2  ./  =  42°  12' 
180x  =  Jl  (42.20) 


Lg  42.20=  1.6253 

Lg  A'  =  0.2268 

-Lgl80=  7.7447 

Lgx  =  0.5968 

X  =  0.3952 

Cliec]<  witli  table. 


As  a  second  example  consider  a  pendnlnm  of  length  a  oscillating  through  an 
arc  of  300°.  Find  the  period,  the  time  when  the  pendulum  is  horizontal,  and  its 
position  after  dropping  for  a  third  of  the  time  required  for  the  whole  descent. 
Let  x'^  4-7/2  =  2  ay  be  the  equation  of  the  path  and  /i  =  a  (l  +  ^  \  3)  the  greatest 
height.  When  y  =  /;,  the  energy  is  wholly  potential  and  equals  myh ;  and  ingy  is 
the  general  value  of  the  potential  energy.    The  kinetic  energy  is 


7?i  /d 
2\c 


m  /ds\"_     I  n 
Kdi)  ~2«7 


dyV 
dtj 


and 


J-WKr 


dij 


y^  \ai/  2  ay  —  y-  \dt 

is  the  equation  of  motion  by  the  principle  of  energy.    Hence 


+  mgy  =  mgh 


Jo    ■ 


ady 


0      -^/'l 


dio 


k"-. 


/gjcit  =  su-i(;(j,  k),         w  =  sn  (Vg/cd,  k),         y  =  hsn~{\ g/at,  k), 


A 

2a 


are  the  integrated  results.  The  quarter  period,  from  highest  to  lowest  point,  is 
K  Va/g ;  the  horizontal  position  is  ?/  =  a,  at  which  t  is  desired  ;  and  the  position 
for  Vg/at  =  §  JC  is  the  third  thing  required. 


A;2  =  0.93301,         2(/ = 


s/k 


K 


1  +  ■\'k 


logf/  = 


-  2  Lg  q' 
i/(l  +  -xk)- 


Lgfc2  =  9.96088 

Lg  Va-  =  9.99247 

V^  =  0.98280 

1-  ^-^  =  0.01720 

1+  ■\^=.  1.98280 


Lg  (l  -  a/a-)  =  8.23553 

Lg(l+  \'A-)  =  9.70272 

-Lg2  =  9.69897 

Lgr/'=  7.63722 

q'  =  0.00434 


Lg2  =  0.3010 

Lg^^'-i  =  0.3734 

-  Lg  3/  =  0.3622 

2Lg(l+  A  A;)  =  9.4034 

L£r  K  =  0.4420. 


Hence  K  =  2.768  and  the  complete  periodic  time  is  4  7i  ^'a/g. 

"  ' / T 

cn  V)  —  V 1  —  a/h,         dn  w  =  a' 1  —  k'-a/h. 


y  =  ^fi 


h 


1     dn!/j        4|4 

— = =  a'  -  k^  =  cot  X, 

^'f^  cn  w        \  3 

Lg  k"  =  9.96988 

Lg  4  =  0.60206 

-Lg3  =  9.52288 

Ltr  cot"  X  =  0.09482 


tan  [TT 

\4 


2  {/  cosh  2  v', 


TT  K      \g   t 
l^\aK 


X  =  43°  26' 12" 
^  TT  -  X  =  1°  33'  48" 
Lg  tan  =  8.43603 
Lg  2  q'  =  9.93825 


Lg  cot  X  =  0.02370         L--  cosh  2v'  =  0.49778 


2/=  1.813 
Lg2/  =  0.2584 
Lg2r/-i  =  9.6266 
Lii'Jf  =  9.6378 


\ 


a  K 


=  9.5228. 


510  THEORY  OF  FUNCTIONS 

llunce  the  lime  for  y  =  a  ii^  t  =  0.3333  K  \  a/g  =  J  whole  time  of  ascent. 

(7  2  ,,      Id       /; /.siiih  7r/v'/3  7v' —  r/'- sinh  7r7v/A''\'- 

y  =  h  811-  -V  '-  -  J'^  \  -  —  ~  [ I 

\  a  3        V  </      k  \cosh  ttK/S  A"  +  ry'^  cosii  ttK/K'J 


n'  ■'  +  '/' ■•  +  '/'-('/'"^  +  n')'  \h'  '■'■  +  ''/='  +  '/ 

iL£rv'  =  0.21241  7'5  =  0.1631  _^    ,    /5.064o\2 

Z/  =  -  "^'     )  • 

-  \  Lgv/'  =  0.78750        q'-  3  =  0.1310  VJ.2003/ 

Tills  gives  y  =  1.732  ((.  which  is  very  near  the  top  at  /i  =  1.866  a.  In  fact  starting 
at  30°  from  the  vertical  the  pendulum  reaches  43°  in  a  third  and  00"  in  another 
third  of  the  total  time  of  descent.  As  sii  \  K  is  (1  +  k')~  ^  it  is  easy  to  calculate 
the  position  of  the  pendulum  at  half  the  total  time  of  descent. 


EXERCISES 

1.   Discuss  these  integrals  by  the  method  of  mapping 

(1^1^ ....  .  .       b 


(a)  z  —■    j  —             —            —  ,  (/>'>>  0.         w  =  h  sn  ciz.       k  =  -  , 

J  0  ^   ^,,-2  _  „.-2^  (//2  _  „,-2j  a 

iii)  z  =    f  "                   '^"'                  ,  w  =  sn2  ('  z.  k).  z  =  2  su-i  (\  j.  k), 

Jo  ^  ((•(!  _  ;/■)  (1  _  k-(i-)  \-        / 


(Iw  sn  (z.  k) 

(1+  ir-^){l  +  k'-hr-i) 

2.  Establish  these  Maclaurin  developments  with  the  aid  of  §  i; 


( V)  .  =    r  "■ '"' .         .•  =  ^"   "   "I  =  tn  (.  k).         z  =  tn-  (.,  k). 

'^'^     ^  (1  +  ir-)  { 1  +  k'-ir-^)  cn  (2.  k) 


(a)    sn  z  =  2  -  (1  +  k-')  j'  +  (1  +  14  f^  +  A-^)  ^ 


(13)  CM  z  =  1  _  -■  +  (1  +  4/,--^)  ~-  -  (1  +  44/,--  +  k;;,-^)  ~  -  +  .  .  . , 
2 :  4 :  (i : 

(7)  dn2  =  l-^--'-  +  k-{i  +  k^)~^-k^(m+  44k-^  +  k^)~  +  .... 
2  !  4  '.  0  I 

3.  I'rove     I      —  —  -    /      —  '  >  1.     siii^^  =  Z^sm^ , 

•^0     ^'l_^-sin-0       ^  '^"      A  1  — /--sin--J/ 


4.   Carr}-  out  tlu'  (•(ini])Utaticiiis  in  tlirsi'  cases  : 


(a)     (       —  to  find  h.          sii-Jv.  /•  (  -  tt.  — —), 

^"     ^  1-  0.1  sin- (9                                -^  \^       ^  10/ 

r'>            <I$                                          1  /I         3    \ 

(/3)     I      —              to  find  7^:,         sn  -  K.  F(  --  tt.  — =  V 

•^0     A  1-  0.0  siii-^                                  3  \3       ^  lo/ 

5.   A  pendulum  oscillates  throuuh  an  an-le  of  (cx)  180%  (;3)  00".  (7)  340°.    Find 

the  periodic  time,  the  posit  inn  at  /  =  j  K.  and  the  time  at  which  the  pendulum 
makes  an  anule  nf  30°  with  the  vertical. 


ELLIPTIC   FUNCTIONS  511 

6.  With  the  aid  of  J>x.  3  find  the  arc  of  the  lemniscate  r-  =  2a'^  coii2  4).  Also 
the  arc  from  0  =  0  to  0  =  30%  and  the  middle  point  of  the  arc. 

7.  A  bead  moves  aronnd  a  vertical  circle.  The  velocity  at  the  top  is  to  the 
velocity  at  the  bottom  as  1  :?i.    Express  the  solntion  in  terms  of  elliptic  functions. 

8.  In  Ex.  7  compute  the  periodic  time  if  )(  =  2,  3,  or  10. 

9.  Neglecting  gravity,  solve  the  problem  of  the  jumping  rope.  Take  the  x-axis 
horizontal  through  the  ends  of  the  rope,  and  the  y-axis  vertical  through  one  end. 
Remember  that  "centrifugal  force""  varies  as  the  distance  from  the  axis  of  rotation. 
The  first  and  second  integrations  uive 


a-tbj  —. ,       /  \  //-  +  <t-j-  h-  —  «' 

ax  —  — z:^^=^=^^^=  .         y  =  ^  lj~  —  II-  sn .     -»   

^  (lyi  _  y2y2  _  a^  \        a-  \  h-  +  a- 


10.  Express     /  —  ,  a  >  1,  in  terms  of  elliptic  functions. 


\  a  —  cos( 

11.  A  ladder  stands  on  a  smooth  tioor  and  rests  at  an  angle  i>f  30^  against  a 
smooth  wall.  Discuss  the  descent  of  the  ladder  after  its  release  from  this  position. 
Find  the  time  which  elapses  before  the  ladder  leaves  the  wall. 

12.  A  rod  is  placed  in  a  smooth  hemispherical  bowl  and  reaches  from  the  bot- 
tom of  the  bowl  to  the  edge.    Find  the  time  of  oscillation  when  the  rod  is  released. 

190.  Legendre's  Integrals  II  and  III.  Tlie  treatment  of 


i 


by  the  method  of  confoi-mal  mujipiug  to  determine  the  fvinetion  and  its 
inverse  does  not  give  satisi'actdvy  results,  for  th(^  map  of  tlie  Iviemann 
surface  on  the  .-.-plane  is  in>t  a  sim]ile  reyion.  J  Jut  the  integral  nuiy  be 
treated  bv  a  change  of  variable  and  be  reduced  to  the  integral  of  an 
elliptic  function.    J-'or  with  (c  =  sn  ii^  n  =  sn~^  //■, 

( 1  —  Irir-)  <hr 


i 


(1  —  /.■-  sn-  if)  du 


0       -\\i-  tr-)(l-]r,r-)       J.  (12) 

=  //  —  /.■"   I       sn-  iiiJii. 

The  problem  thus  becomes  that  of  integrating  sn"  u.    To  effec-t  the  in- 
tegration, sn-  II  will  be  expivssed  as  a  derivative. 

The   function   sn- //   is  doubly  })eriodie  with  })eriods  2  K,  2  IK',  and 
with  a  pole  of  the  second  order  at  //  =  iK'.    But  now 

Q(ii  +  2  A')  =  ©(//),  ©(^  +  2  lK')  =  -q-'^e"^"Q(u) 


ITT 


log  ©(^^  -f  2  K)  =  log  ©(/'),  log  (©  +  2  IK')  =  log0(/^)  -  —  If  -  log  (-  q). 


K 


512  THEORY  OF  FUNCTIONS 

It  then  appears  that  the  second  derivative  of  log  @(ii)  also  has  the 
periods  2  K,  2  iK'.    Introduce  the  zeta  function 

ZOO  =  -flog  000  =  ^'  Z'('0  =  f  ®^.  (13) 

The  expansion  of  ©'(?0  shows  that  0'(?O  =  0  at  u  =  mK.  About  u  =  Hi' 
the  expansions  of  Z'(?0  and  &n^u  are 

^'^")  =  -0^-:^^  +  ^'o  +  ---.       «"^-  =  |(.-^y.')^  +  ^o  +  ---- 

Hence  A;^  sn^  7f  =  -  Z'('0  +  ^'(0),         Z'(0)  =  0"(O)/0(O), 

and  Jr   f    sn'^n  die  =  -  Z(;0  +  ''^'(O), 

(1  -  /.:-  sn-  u)  du  =  u(l-  Z'(0))  +  Z  (h).  (14) 


Jo 


The  derivation  of  the  expansions  of  7,'{u)  and  sn^  u  about  m  =  IK'  are  easy. 

e(u)  =   C-rT(l  -  r/2«+lc^A-U  loge(M)  =  ^  l0g(l  -  q-2n+le^K")  +  log  C 

(         -'"-A 

log  6  (w)  =  log  \1  —  qc    '^   )  -{■  function  analytic  near  u  =  iK'. 

in 


K{l-(ie    A-   j  Jvl^e^^'    -q) 

in 

f{u)  ^  cx  "  =f{iK')  +  («  -  iK')r{iK')  +  .  .  .  =  ry  +  (,,  _  iZr)  ^9  + 

e'(u)  _     +  1  (Z  e'(M)  _      - 1 

Q{u)       u  —  iK'            '  du  Q(u)       (u  —  iK')'^ 

sn  (w  +  jA'')  = ,  sn2(w  +  fJv')  =: , 

k  sn  M  k-  sn'^  u 


/(u)  =  sn  u  =  uf{0)  +  I  u^r"{0)  +...  =  u  +  cu^  +  •  •  • , 

sn2(u  +  iK')  =  —        -  =  ^  ( CU+  ■  ■  ■]  =  -  (—  —  2c  + 

A;^  sn'^  u      k-  \u  /       t-  \m'- 

sn-'  M  =  —  I 2c  +  ■ 

k-^  \{u  -  i/iT')"'^ 

In  a  similar  manner  may  be  treated  the  integral 

die  r"        dn 


■)• 


Ji) 


Jo      (?/;'  -  a)  V(l  -  vr'-^)  (1  -  J^^ic^       Jo      ^J^'  "  "  « 
Let  a  be  so  chosen  that  sn-  ft  =  (x.    The  integral  becomes 


(III) 


—. ^  =  7, , —    I    -, ^ — ^^«-     (15) 

g     sn-  K  —  sn  a       J  sn  «  en  a  dn  f(  J       sn-  u  —  an  a  ' 


ELLIPTIC  FUNCTIOXS  513 

The  integrand  is  a  function  with  periods  2  A',  2  IK'  and  with  simple 
poles  at  ;(  =  -|-  r^    To  find  the  residues  at  these  poles  note 

,.                H^<1  -.  1  +1 

lim   — :, r;—  =  lim 


,  „  sn'-  it  —  sn'^  a     u  =  ±a  2  sn  u.  en  u  dn  u       2  sn  a  en  (t  dn  (t. 

The  coefficient  of  (i/  =f  a)~'^  in  expanding  about  ±  a  is  therefore  ±  1. 
Such  a  function  niay  be  written  down.    In  fact 

2  sn  a  en  a  dn  a  _  H'(u  —  a)       //'(w  -\-  a) 
sii^  u  —  sn'-a  II  (^ti  —  c/)       //(w  +  f) 

=  z,(«  -  ^0  -  ZiC''  +  ^0  +  ^'' 

if  Zj  =  IV I II.    The  verification  is  as  al)Ove.    To  determine  ('  let  ?<.  =  0. 

2cnr/  dn^/       ^      ,  ,      ,  1     //('') 

Then         C  = h  2  Z,(r/),    but     sn  ?<  =  ^-  7-^  ^ 

sn  (/  '^  ^  V/.-  ®(") 

and  -7-  log  sn  ?^  = =  Z.iin  —  Z(u). 

(In  sn  u  '^   •^  ^  ^ 


Hence  C  reduces  to  2Z('')  and  the  integral  is 
du  1 


f 


sn-  u  —  sn-  a       2  sn  a  en  «  dn  ^/ 


(16) 


The  integrals  liere  treated  by  the  substitution  w  =  sn  u  and  thus  reduced  to  the 
integrals  of  elliptic  functions  are  but  special  cases  of  the  integration  of  any  rational 
function  A'(it',  V  U')  of  V3  and  the  radical  of  the  biquadratic  W  =  (1  —  vfl){\  —  U-w^). 
The  use  of  the  substitution  is  analogous  to  the  use  of  lo  =  sin  «  in  converting  an 
integral  of  A'(k',  a  1  —  ic-)  into  an  integral  of  trigonometric  functions.  Any  ra- 
tional function  A' (('.•,  ^   IT)  niay  be  written,  by  rationalization,  as 

i?(w,VTr)  ^  ^M±-^l(i-ll^  =  AOo)  +  fi(i»)Vlr 

where  U  means  not  always  the  same  function.  The  integral  of  7?(h',  •vMT)  is 
thus  reduced  to  the  integral  of  i?^(?t')  which  is  a  rational  fraction,  plus  the  inte- 
gral of  xvRJ^w"-)/^^'  which  by  the  substitution  vfl  —  u  reduces  to  an  integral  of 
li  (u.  \  (1  —  i()(l  —  i'-M)  and  may  be  considered  as  belonging  to  elementary  calculus, 
l>lus  finally 

rIL{iv-)  ,  r 

I      " -dv'  =  I  RJsn-u)du,        w  —  snu. 

By  the  method  of  partial  fractions  li.^  may  be  resolved  and 

/p         du 
sn- "  u  du         71  S  0,  I  71  >  0 

J   (sn-  u  —  a)" 

are  the  types  of  integrals  which  must  be  evaluated  to  finish  the  integration  of  the 
given  U(ii\  \^]V).    An  integration  by  parts  (B.  O.  Peirce,  No.  507)  shows  that  for 


514  THEORY   OF  FUNCTIONS 

the  first  type  n  may  be  lowered  if  positive  and  raised  if  negative  until  the  integral 
is  expressed  in  terms  of  the  integrals  of  sn-^x  and  sn°  x  =  1,  of  which  the  first  is 
integrated  above.  The  second  type  for  any  value  of  n  may  be  obtained  from  the 
integral  for  ?i  =  1  given  above  by  differentiating  with  respect  to  a  under_the  sign 
of  integration.  Hence  the  whole  problem  of  the  integration  of  Ii{w,  V  W)  may 
be  regarded  as  solved. 

191.   With  tlie  substitution  v  =  sin  <j>,  the  integral  II  becomes 


Vl  -  //-  sin-  OJO  =   /  '"  (Iw  (^j^ 

=  u  (1  -  Z'(0);)  +  Z  (v),  „  =  F{<i>,  /.•). 

Ill  purticulur  /■>'(  .\  tt,  /.•)  is  caUed  the  eomph'te  integral  of  the  second  kind 
and  is  generally  denoted  l)v  L'.  ^Vhen  cf)  =  \  tt,  the  integral  n  =  /'X^,  A") 
becomes  the  complete  integral  A'.    Then 

E  =  K  (1  -  Z'(0))  +  Z  {K)  =  K  (1  -  Z'(Oj),  (18) 

and  E(ci>,  /.•)  =  EF(cf>,  k)/K  +  Z(//).  (19) 

The  problem  of  computing  E(<^.  /.■)  thus  reduces  to  that  of  computing 
K,  E,  E{(^,  /.•)  =  }i,  and  Z{ti).  The  methods  of  olitaiuing  A'  and  l-^i^.  /.') 
liave  been  given.  The  series  for  Z(ii)  converges  ra})idly.  The  value 
of  A'  may  be  found  by  computing  A'(l  —  Z'(0)). 

For  the  convenience  of  logarithmic  com})utatioii  note  that 


K—E  ,^^^.        0"(O)  TT        2  7r%  ,    ,       ^    . 

or  A-- /;=  i-Tr/VZ^ -(2  77/ A-)'^y  (1-4  y^ +  ■••).  (20) 

M  -7/   X      Q'^'O       2  yTT         sin  2  V  -  2  y'^  sin  4  v  H ,^... 

Also         Z(?0  =  — ~~"  = :. ; : 7r~^ \ "  (21) 

■   '       0  (//)  A     1  —  2  y  cos  2  V  +  2  y-*  cos  4  v ^     ' 

wliere  v  =  ^n /2  K.  These  series  neglect  only  terms  in  y'',  -which  will 
barely  atfect  the  fifth  place  when  /.•  ^  sin  82°  or  /r  ^  0.98.  The  series 
as  Avritteii  therefoin^  cover  most  of  the  cases  arising  in  practice.  For  in- 
stance in  the  iirohh'Ui  which  gives  the  name  to  the  elliptic  functions 
and  integrals,  the  problem  of  hndiiig  tlie  arc  of  the  ellipse  x  =  a  sin  <^, 

y  =  /;  cos  (^. 

r/.s'  =  V""  COS-  ^  +  //'-  sin'-  <^il(^  =  (I  V  1  —  ('-  sin"-  <^(1<^  ; 

the  eccentricity  r  may  be  as  liigli  as  0.99  without  invalidating  the 
approximate   formulas.    An  exam})le   follows. 

Let  it  be  required  to  detenniiic  tlie  length  of  the  quadrant  of  an  ellipse  of 
eccentricity  e  =  0.',»  and  also  tlic  Icimth  df  the  portion  over  half  the  seniiaxis 
major.    Here  the  series  in  y'  converge  better  tlian  those  in  y.  but  as  the  proper 


ELLIPTIC  FUNCTIONS  515 

expression  to  replace  Z(h)  has  not  been  found,  it  will  be  more  convenient  to  use 
the  series  in  q  and  take  an  a(lditi(jnal  term  or  two.    .\s  k  —  ().'.>.  k"-  =  0.1!). 

Lgr-  =  9.27875  Lg(l  -  VP)  =  9.53120  5dilf.  =  G. 55515 

Lg  V A?  =  9.81909  Lg (l  +  VP)  =  0.2201 7  Lg  10  =  1 .20412 

VF  =  0.66022  diff.  =  9.31 103  Lg  term  2  =  5.35103 

1  -  VP  =  0.33978  Lg  2  =  0.30103  term  1  =  0. 102.33 

1  +  vF  =  1.66022  Lg  term  1  =  9.01000  term  2  =  0.00002 

q  =  0.10235. 

Lg  g  =  9.0101  Lg  2  TT  =  0. 7982  Lg  J  tt/ VP  =  0.3764 

3  Lg  r/  =  7.0303  -  2  Lg  (l  +  ^  k')  =  9.5597  |  log  2  -rr/K  =  0.6003 

4Lgg  =  6.0404  Lg(l+ 2  g-*)  =0.0001  Lgr/  =  9.0101 

^3  =  0.0011  Lg  K  =  0.3580  Lg  (1-4  r/S)  =  9.9981 

(/  =  0.0001  A'  =  2.280  Lg  (A'  -  A)  =  0.0449. 

Hence  A— A  =  1.109  and  A=  1.171.  The  (jnadrant  is  1.171  r<.  The  point  cor- 
responding to  X  =  A  a  is  given 

LgdnA=  9.9509 
Lg  VP  =  9.8197 
LgcotX  =  0.1312 
X  =  36^  28^' 

Now  180  F  =  A  (42.92).  The  computation  for  A,  Z,  A(i  tt)  is  then 

Lg  K  =  0.3580  Lg  2  tt/A  =  0.4402           Lg  A/ A  =  9.7106 

Lg  42.92  =  1.6326  L^  7  =  9.0101                 Lg  A  =  9.7353 

-  Lg  180  =  7.7447  Lg  sin  2  v  =  9.8331              AA/A  =  0.2792 

Lg  A  =  9. 7353  -  Lg  (1  -  2  ry  cos  2  p)  =  0.0705                        Z  =  0.2256  * 

A  =  0.5436  Lg  Z  =  9.3539             A  (i  tt)  =  0.5048. 

The  value  of  Z  marked  *  is  corrected  for  the  term  —  'Iq^nin  iv.  The  part  of  the 
quadrant  over  the  first  half  of  the  axis  is  therefore  0.5048  a  and  0.666  a  over  the 
second  half.  To  insure  complete  four-figure  accuracy  in  the  result,  five  places 
should  have  been  carried  in  the  wcjrk,  but  the  values  here  found  check  with  the 
table  except  for  one  or  two  units  in  the  la.st  place. 

EXERCISES 

1.  Prove  the  following  relations  for  Z(m)  and  7.^(u). 
Z  (-  M)  =  -  Z  («),         Z  (u  +  •2K)  =  Z  (h),         Z{u  +  2  iK')  =  Z  (»)  -  ?7r/A. 

If  Z,{u)  =  ~  log  II  («)  =  ^[^  ,         Z,{u  4-  iK')  =  Z  (.)  -  —  , 

du  II  {u)  2K 


.  =  30^ 

Then  dn 

A 

=  ^  1  —  0.2025. 

Ltt-X 

=  8^31.r 

cos  2^  =  0.7323 

Lgtan 

=  9.1758 

Hence     4  v  near  90^ 

Lg  2  q 

=  9.3111 

14-  2  7*cos4»'=  1.0000 

f  cos  2  V 

=  9.8647 

2  J/  =  42=  55'. 

Z[(u)  +  Z'(0),  r  -^  =  -  Z,(»)  +  wZ'(O), 

J    sn-  u 


1  „.,  ,  .  r  du 

sn"^  u 


ZAu)  —  Z  (m)  =  ~  log  sn  u  = ,         Z,  (0)  =  00. 

du  su  u 


516  THEORY  OF  FUNCTIONS 

2.  All  elliptic  function  with  periods  2  A',  2  iK'  and  simple  poles  at  «j,  Qo,  ■  ■  • ,  On 
with  residues  Cj,  c.,,  •  •  • ,  c„,  2c  =  0.  may  be  written 

f{u)  =  CiZi(w  —  a,)  +  c.,Zi(«  -  «,)  +  ■  •  •  +  '-»Z,(«  —  a„)  +  const. 

_    A;2  sn  a  en  a  dn  «  sn'^  u       1,  ,       1„,     ,     \   ,   „,.  s 

3-  —, TTy — o S =  -  Z  («  -  a)  -  -  Z  (u  +  a)  +  Z'(a), 

1  —  A;''  sn^asii-  u  2  2 

r"         sn'-^wdn  1,      e(a  — k)         „,,  , 

A;"^  sn  a  en  a  dn  a   |      ^ — ; —  =  -loir-— ^ --\-u7.{a). 

J{)     1  —  k-  811-  a  sn'-^  w       2        9  {«  +  i() 


^^"           ^    ^„,A^          \v/    \'  ^           -cn^X»dn^X« 
—  =  X((Z  (0)  —  ^  \Z  (\  X»j  —  ^  X ^ 

sii-  \  \u  sii  A  \u 


-^,  /           .      ,           -  X          -  en  \  Xi(  (111  ^  \u       „ 
=  Xi(  —  ^  X£  (0  =  siii-'sii  \  X»)  —  A  X ; h  C, 

811  \  X« 

(B)     I =    I   (111- i/(Z«  —  ;^'- =  i:((;6  =  8111-1  sii  «)  —  A-- ^ 

J    (In-  u       J  dn  u  dn  u 

,    ,     /•     en- m'7h  ^  ,, ,  .      ,         ,  cii  h      ,,       ^  ,   „    v 

(7)     I =  u  —  •2L{<p  =  8111-1  ^1,  „)  .|. (1  —  2  dn^  «). 

J    811-  H  dn-  «  811  u  dn  (; 

5.  Find  the  length  of  the  quadrant  and  of  the  portion  of  it  cut  off  by  the  latus 
rectum  in  ellipses  of  eccentricity  e  =  0.1,  O.o,  0.75,  O.U'). 

6.  If  e  is  the  eccentricity  of  the  hyperbola  .c-/a-  —  !/'-/l>-  =  1,  sIkjw  that 

b'    r  *       sec''^  <pdd>  ,         ae  ,       ' 

s  =  —   I      — —  ,         where  —  u  =  tan  cp,         k 

aeJo    Vi_/^-^sin2<*  ^- 


e 


Ij^ 


=  —  F((p.  k)  —  acE{(p,  k)  +  octaii  (p  \  I  —  k-tiin-<p. 
ue 

7.  Find  the  arc  of  the  hyperb(.)la  cut  off  by  the  latus  rci.;tuiii  if  e  =  1.2,  2,  3. 

8.  Show  that  the  length  of  the  jumping  rope  (Ex.  0.  p.  .311)  is 

a(k'K/\  2  +  ^  2  E/k'). 

9.  A  flexible  trough  is  filled  with  water.    Find  the  expression  of  the  shape  of 
a  cross  section  of  the  trough  in  terms  of  F{4>.  k)  and  E {(p,  k). 


10.  If  an  ellipsoid  has  the  axes  a  >  h  >  c.  lind  the  area  of  one  octant. 

1    ,     Tzdh  V c-  ^^ ,    ,^    it- —  (•- ,,,    ,,n  '• 

,  TTi-  +  -  -. —      -„  /•  {4,,   k)  + /•;  (</),  /.•)    ,         c(  IS  0  =  ^  ,         A-- 

4  4  sill  0  L""  ""  J  « 

11.  Compute  the  area  of  the  ellipsoid  with  axes  3,  2,  1. 


h-^  -  <■■■ 
^- sill-  ( 


12.  A  hole  of  radius  b  is  bored  through  a  cylinder  of  radius  a>b  centrally  and 
perpendicularly  to  the  axis.    Find  the  volume  cut  out. 

13.  Find  the  area  of  a  right  elliptic  cone,  and  compute  the  area  if  the  altitude 
is  3  and  the  semiaxes  of  the  base  are  \\  and  1. 


ELLIPTIC  rUNCTIOXS  517 

192.  Weierstrass's  integral  and  its  inversion.  In  studying  the 
general  theory  of  doubly  periodic  functions  (§  182),  the  two  special 
functions  ^^(^/),  p'(i')  were  constructed  and  discussed.    It  was  seen  that 


=  I  ='         e^  +  r, +  .'  =0, 


where  the  fixed  limit  x  has  l)een  added  to  the  integral  to  make  w  =  cc 
and  .-;  =  0  correspond  and  where  the  roots  liave  been  called  e^,  e_^,  e.^ 
Conversely  this  integral  could  be  studied  in  detail  by  the  method  of 
mapping ;  but  the  method  to  be  followed  is  to  make  only  cursory  use 
of  the  conformal  maj)  sufficient  to  give  a  hint  as  to  liow  the  function 
p('')  iiiay  be  expressed  in  terms  of  the  functions  sn  ,~  and  en-.-.    The 

discussion  will  be  restricted  to  the        

case  which  arises  in  practice,  namely,     w,  _i,     ^         ^-^H-.        -^-l"— 

when  y.,  and  r/^  are  real  quantities.  2'     2       -oo'  <C^        ^i      +°= 

There  are  two  cases  to  consider,  one 

when  all  three  roots  are  real,  the  other  when  one  is  real  and  the  other 
two  are  conjugate  imaginary.  The  root  e^  will  be  taken  as  the  largest 
real  root,  and  e,-,  as  tlie  smallest  root  if  all  three  are  real.  Xote  that  the 
sum  of  the  three  is  zero. 

In  the  case  of  three  real  roots  the  Eiemann  surface  may  be  drawn 
with  junction  lines  c,^.  f.,,  and  f^,  x.  The  details  of  the  map  may  readily 
be  filled  in,  but  tlie  observation  is  sufficient  that  there  are  oidy  two 
essentially  different  paths  closed  on  the  surface,  namely,  about  e.,,  e^ 
(which  by  deformation  is  equivalent  to  one  about  e^,  cc)  and  al)out  f^,  e^ 
(which  is  equivalent  to  one  about  f,,,  —  x).  The  integral  about  e,-,,  e^  is 
real  and  will  be  denoted  by  2  w^,  that  about  e.^,  e^  is  pure  imaginary  and 
will  be  denoted  by  2  w.,.  If  the  function  yv  (.t)  be  constructed  as  in  §  182 
with  w  =  2  Wj.  w'  =  2  (0.,  the  function  will  have  as  always  a  double  pole 
at  ,~  =  0.  As  the  })eriods  are  real  and  pure  imaginary,  it  is  natural  to 
try  to  exjiress  y/ ('.')  in  terms  of  sn  z.  As  y;  (-')  depends  on  two  constants 
f/.,,  y.j,  "wherras  sn  ,-.'  depends  on  oidy  the  one  /.-,  the  function  y/ (.-r)  will 
be  exjiressed  in  terms  of  sn  (  VA::,  /.•),•  where  the  two  constants  A,  /.•  are 
to  be  determined  so  as  to  fulfill  the  identity  j/-  =  -iji^  —  f/„j>  —  y^.  In 
particular  try 

2j('S)  =  A  H — J         .1,  X,  /.■  constants. 

sn-(-vA.v,  k) 


518 


THEORY  OF  FUNCTIONS 


This  form  surely  gives  a  double  pole  at  z  =  0  with  the  expansion  \J^. 
The  determination  is  relegated  to  the  small  text.    The  result  is 


i>(^)  =  .,+ 


h^ 


e.,  —  p.. 


-<1, 


(23) 


sn'^  (  VXz,  Jc) 
X  =  «j  —  ^2  >  0,  coj  VX  =  A',  o)^  Va  =  iK'. 

In  the  case  of  one  real  and  two  conjugate  imaginary  roots,  the 
Riemann  surface  may  be  drawn  in  a  similar  manner.  There  are  again 
two  independent  closed  paths,  one  about  e.„  e.^  and  another  about  e.^,  e^. 
Let  the  integrals  about  these  paths  be  respectively  2  w^  and  2  w.,.   That 


2  coj  is  real  may  be  seen  by  deforming  the  path  until  it  consists  of  a 
very  distant  portion  along  which  the  integral  is  infinitesimal  and  a  path 
in  and  out  along  e.^,cx),  which  gives  a  real  value  to  the  integral.  As 
2  w,^  is  not  known  to  l)e  pure  imaginary  and  may  indeed  be  shown  to  be 
complex,  it  is  natural  to  try  to  express  j^i.'-)  i^^  terms  of  en  z  of  which 
one  period  is  real  and  the  other  complex.   Try 

l  +  (m{2^z,  k) 


p(z)^A  +/X 


l-cn(2  V/x,t;,  k) 


This  form  surely  gives  a  double  pole  at  «  =  0  with  the  expansion  1/?, 
The  determination  is  relegated  to  the  small  text.    The  result  is 


P  (^  =  '\  +  f^ 


l,  +  cn(2V/.^,  /•) 


k' 


l-cn(2  V/xs, /v)  ^ 

/*'  =  (''i  -  'QO'i  -  ^'3)'  ^f^^i  =  ^J  ^f^<^ 

To  verify  these  determinations,  substitute  in  2/'^  =  ip^ 


4/x 


(23') 


p  {z)  =  A  + 


sn2(VX2,  k) 


l>\z)  =  ■ 


2\2 


'(VX2,  A:) 


en  (  vXz,  A;)  dn  (a/xz,  k\ 


4X3 


(1-  sn2)(l-  fc2sn2) 


4|yl3  + :r+^,  -  +  ^ 

sn-  sn'*         sn" 


-  o-A 


f/oX 


-  9z  ■ 


Eijuate  coefficients  of  corresponding  powers  of  sn'-.    Hence  the  equations 

4  ^13  -  g„A  -  (/3  =  0,         4  X2A;2  =  12  yl2  _  y.,\         -  X  (1  +  fc2)  =  3  J.. 


ELLIPTIC  FUNCTIONS 


519 


The  first  shows  that  ^  is  a  root  e.    Let  A  =  e.^.    Note  —  g.-,  —  e^e^  +  e^e^  +  ^-fi-i- 
\ .  \k~  =  3  e.,-  +  e^fo  +  e^c^  +  c.-,e.^  =  (e^  —  e.-^{i;.^  —  e.,). 
\  -\-  \k'-  =  —  3  e,  =  t'^  —  <?2  +  ^V,  ~  ^-j' 
by  virtue  of  tlie  relation  e^  +  e.,  +  e^  =  0.    The  sohuiou  i.s  iiiunediate  as  given. 
To  verify  the  second  determination,  the  substitution  is  similar. 
l  +  cn2V,u2  „_^  4^i2sndn 


p{z)  =A+fM 


[i/(^)]-^  =  l(5M^ 


1  —  en  2  V/xz 
(1  +  cn)(k'-  +  k-cn-) 


PV) 

(1  -  en)- 

■ilj.^[t^  +  2{l-2k-)f^  +  t] 


(1  -  en) 

where  i  =  (1  +  cn)/(l  —  en).    The  identity  -p"-  =  -ip'-^  —  g.,p  —  g.,  is  therefore 
if,S[L^  +  2{\-  2  k-)  r-  +  t]=-i:  (.43  +  3 .4-V  +  3  Af.f^  +  ^0)  -g.^A  -  g.^,xt  -  g^. 
•4.43- f/._,.4 -r/..  =0,         4  m- =  12.4 --(/,.         2  ^  (1  -  2 /,•-)=  3.4. 
Here  let  .4  =  tj.     The  .solution  then  appears  at  once  from  tlii'  forms 

AC-  =  3 e^-  +  e,c,  +  e,e,  +  (-.,63  =  (c,  -  e;)(e,  -  c,).         ix{\  -  2 k-)  =  3 .4/2. 

The  expression  of  the  function  j)  in  ternts  of  the  functions  ah'eady 
studied  permits  the  determination  of  the  vahu'  of  the  function,  and  by 
inversion  permits  tlie  sokttion  of  the  equation  jj  (z)  =  c.  Tlie  function 
p{z)  may  readily  be  expressed  directly  in  terms  of  the  theta  series. 
In  fact  the  periodic  properties  of  the  function  and  the  corresponding- 
properties  of  the  quotients  of  theta  series  allow  such  a  representation 

Qgj 20]+2a32  looi  2(i5i+26ao 


S^:.^ 


?~i.2 


0  -co<p<0        o<p<as      2(i3i      0 -co  <p'<o     o<p'<a:>    2ft3, 

to  be  made  from  the  work  of  §  175,  provided  the  series  be  allowed  com- 
plex values  for  q.  l>ut  for  practical  purposes  it  is  desirable  to  liave  the 
expression  in  terms  of  real  quantities  only,  and  this  is  the  reason  for  a 
different  expression  in  the  two  different  cases  here  treated.* 

The  values  of  z  for  which7^(.-;)  is  real  may  be  read  off  from  (2o)  and 
(23')  or  from  the  correspondence  between  the  yr-surface  and  the  ,t'-})lane. 
They  are  indicated  on  the  figures.  The  functions  j>  and/>'  may  be  used 
to  ex})ress  parametrically  the  curve 

4  ,-'  -  y.,7-  -  ff^     by     t/  =  jj'(z),         X  =  p  (,-)• 


1        "" 

a               2- 

•2 

^  e3>p>e„ 

=,    P'<o 

/    Q  1    ^ 

^'e     / 

/     lis 

vis     / 

/     '^i-^ 

a^i-r  / 

/        ^,"?s 

VI  ?Y 

/            '^l 

81  / 

/oo>p>ei  j  ei<p<co  '  1/ 

//' 


*  It  is.  however,  jjossihle,  if  desired,  to  transform  the  ifiven  cubic  4  f"^  —  g.,>'-  —  g^  with 
two  cdiiiplex  roots  into  a  similar  euhie  with  all  three  roots  I'eal  and  tiie.s  a\ciid  tlie  din)li- 
cate  forms.    41ie  trausfonuatiou  is  nut  <riveu  here. 


520 


THEORY  OF  FUXCTIONS 


2=w„-t-tt 


The  figures  indicate  in  the  two  eases  the  shape  of  the  curves  and  the 
range  of  values  of  the  parameter.  As  the  function  p  is  of  the  second 
order,  tlie  equation  |>  (,-j)  =  c  has  just  two  roots  in  the  parallelogram, 
and  as  jji?-^  is  an  even  function,  they  will  be  of  the  form  z  =  a  and 
2;  =  2  Wj  -f-  2  w.^  —  a  and  l)e  symmetri- 
cally situated  with  respect  to  the  cen- 
ter of  the  figure  except  in  case  a  lies 
on  the  sides  of  the  parallelogram  so 
that  2  o)j  +  2  w,  —  a  would  lie  on  one 
of  the  excluded  sides.  The  value  of 
the  odd  function  ^>'  at  these  two  points 
is  equal  and  opposite.  This  corresponds  precisely  to  the  fact  that  to 
one  value  x  =  c  of  x  there  are  two  equal  and  oi)posite  values  of  //  on 
the  curve  \f-  =  4  x'^  —  rj,^x  —  g,^.  Conversely  to  each  point  of  the  }jarallelo- 
gram  corresponds  one  point  of  the  curve  and  to  points  symmetrically 
situated  with  respect  to  the  center  correspond  points  of  the  curve  sym- 
metrically situated  with  respect  to  the  a^-'axis.  Unless  z  is  such  as  to 
make  both^>(s:)  and2>'(s)  real,  the  point  on  the  ciirve  will  he  imaginary. 

193.  The  curve  y'^  =  \x^  —  (j„x  —  {/.,  may  be  studied  by  means  of  the  properties 
of  doubly  periodic  functions.    For  instance 

Ax  +  liy  Ar  V  -  Ap'{z)  +  Bp{z)  +  C  =  0 

is  the  condition  tliat  tlie  parameter  z  sliould  be  such  that  its  representative  point 
shall  lie  on  the  line  Ax  +  By  +  C  =  0.  But  the  function  AjViz)  +  Bp{z)  +  C  is 
doubly  periodic  with  a  pole  of  the  third  order  ;  the  function  is  therefore  of  the 
third  order  and  there  are  just  three  ^joints  Zj,  z.,,  z.3  in  the  parallelogram  for  which 
the  function  vanishes.  These  values  of  z  correspond  to  the  three  intersections  of 
the  line  with  the  cubic  curve.  Now  the  roots  of  the  doubly  periodic  function  sat- 
isfy the  relation 

Zj  +  Zo  +  Zo  —  3x0  =  2  m^w^  +  2  m.,w.y 

It,  may  be  observed  that  neither  m^  nor  )/).,  can  be  as  ureat  as  .3.  If  conversely  Zj.  z.,,  z^ 
are  three  values  of  z  which  satisfy  the  relation  Zj  -\-  z„  -|-  Z;,  =  2  m^ij]^  +  2?n.,w.,.  the 
three  corresponding  points  of  the  cubic  will  lie  on  a  line.  F(jr  if  Zg  be  the  point  in 
which  a  line  thrtnigh  z,,  z.,  cuts  the  curve, 

Zj  +  z.i  +  z'3  =  2  ///jojj  +  2  MfoO).,.         Zg  —  Zj  =  2  (»(j  —  ?/ij)  a)j  4-  2  ()«._,  —  m'.^  w.,, 

and  hence  z.,.  z^  are  identical  except  for  the  addition  of  periods  and  nuist  therefore 
be  the  same  point  on  the  parallelogram. 

(»ne  application  of  this  condition  is  to  find  the  tangents  to  the  curve  from  any 
point  of  the  curve.  Let  z  Ijc  the  point  from  which  and  z'  that  to  which  the  tangent 
is  drawn.   The  condition  then  is  z  +  2z'  =  2  ))i^w^  +  2  in.,w.,.  and  hence 

z'  =  —  Iz,  z'  =  —  J  Z  +  a)[ .  z'  =  —  I  Z  -\-  w., .  z'  =  —  ^  Z  +  a>,  +  o)., 

are  the  four  different  possibilities  fur  z'  corres})onding  to  nt^  =  ;/(.,  =  0  :  ?/(,  =  1, 
nt.,  =  0  ;   //(.  =  0,  m.,  =  1  ;  in.  =-  1,  ;//.,  =  1.    To  give  other  values  to  ))i^  or  ///„  wou'ld 


ELLIPTIC  FUXCTIOXS  621 

merely  reproduce  one  of  the  four  points  excejjt  for  the  addition  of  complete  periods. 
Hence  there  are  four  tangents  to  the  curve  from  any  point  of  the  curve.  The 
question  of  the  reality  of  these  tangents  may  readily  be  treated.  Suppose  z  denotes 
a  real  point  of  the  curve.  If  the  point  lies  on  the  infinite  portion,  0  <  z  <  2  w^,  and 
the  first  two  points  z'  will  also  satisfy  the  conditions  0  <  z'  <  2  a;j  except  for  the 
possible  addition  of  2u;j.  Ilcnce  there  are  always  two  real  tangents  to  the  curve 
from  any  point  of  the  infinite  branch.  In  case  the  roots  gj,  €„,  Cg  are  all  real,  the 
last  two  points  z'  will  correspond  to  real  points  of  the  oval  portion  and  all  four 
tangents  are  real  ;  in  the  case  of  two  imaginary  roots  these  values  of  z'  give  imag- 
inary points  of  the  curve  and  there  are  only  two  real  tangents.  If  the  three  roots 
are  real  and  z  corresponds  to  a  point  of  the  oval,  z  is  of  the  form  w.,  +  »  and  all 
four  values  of  z'  are  complex, 

—    \  (i>.,  —   I  U,  —   I  Wo  —    I  U  +   O),,  +    J  W.,  —   I  «,  +    I  OJ.,  —   III  +    OJj, 

and  none  of  the  tangents  can  be  real.   The  discussion  is  complete. 

As  an  inflection  point  is  a  point  at  which  a  line  may  cut  a  curve  in  three  coin- 
cident points,  the  condition  3z  =  2??i,Wj  +  2??i,,aj.,  holds  for  the  parameter  z  of  such 
points.    The  possible  different  combinations  for  z  are  nine  : 

z  =  0  I  w.,  i  0,., 

f  wj         I  ojj  +  I  w.,         f  u);  +  I  w., 

i^l  |Wi+|Wo  ^O),   +   ^w.,. 

Of  these  nine  inflections  only  the  three  in  the  first  colunm  are  real.  When  any 
two  inflections  are  given  a  third  can  be  found  so  that  Zj  +  z.,  -f-  Zg  is  a  complete 
period,  and  hence  the  inflections  lie  three  bj^  three  on  twelve  lines. 

If  p  and  ]/  be  sul)stituted  in  Ax'~  +  lixy  +  Cij'^  +  l)x  +  Ey  +  l'\  the  resuH  is  a 
doubly  periodic  function  of  order  0  with  a  pole  of  the  0th  order  at  the  origin. 
The  function  then  has  6  zeros  in  the  parallelogram  coiuiected  by  the  relation 

Zj  +  z.,  +  z.^  +  z^  +  Z.  +  z^  =  2  ?/i,c<;,  -1-  2  m.,w.,, 

and  this  is  the  condition  wliich  connects  the  parameters  of  the  0  points  in  which 
the  cubic  is  cut  by  the  conic  Ax^  +  Bxy  +  Cy-  -f  Dx  +  Ey  -[-  F=  0.  One  applica- 
tion of  interest  is  to  the  discussion  of  the  conies  which  may  be  tangent  to  the  cubic  at 
three  points  z,.  ?.,.  z.,.  'i'lie  condition  then  reduces  to  Zj  +  z.,  +  z.,  =  jh^w,  +  ?n.„w„. 
If  ??ij,  m.^  are  0  (ir  any  even  number.s,  this  condition  expresses  tlie  fact  that  the 
three  points  lie  on  a  line  and  is  thei'efore  of  little  interest.  'J'he  other  possibilities, 
apart  from  the  addition  of  complete  periods,  are 

z^  4-  Zo  +  Zo  =  Wj,         Zi  -f-  z.-,  +  Z3  =  w.,,         z^  +  z.,  +  z.,  =  wj  +  Wo. 

In  any  of  the  three  cases  two  points  may  be  chosen  at  random  on  the  cubic  and 
the  third  point  is  then  fixed.  Hence  there  are  tiiree  conies  which  are  tangent  to 
the  cubic  at  any  two  assigned  points  and  at  some  other  point.  Another  application 
of  interest  is  to  the  conies  which  have  contact  of  the  5th  order  with  the  cubic. 
The  condition  is  then  Gz  =  2?/),w,  +  2in.,o}.,.  As  ?Hj,  ?Ho  niay  have  any  of  the  0 
values  from  0  to  5,  there  are  30  points  on  the  cubic  at  which  a  conic  may  have 
contact  of  the  5th  order.  Among  these  points,  however,  are  the  nine  inflections 
obtained  by  giving  m^,  m.-,  even  values,  and  these  are  of  little  interest  because  the 
conic  reduces  to  the  inflectional  tangent  taken  twice.  There  remain  27  points  at 
which  a  conic  may  have  contact  of  the  5th  order  with  the  cubic. 


522  THEORY   OF  FUNCTIONS 

EXERCISES 

1.  The  function  f (2)  is  defined  by  the  equation 

-^'{z)=p{z)     or     ^{z)  =  -  fp{z)dz  =  ~--c^z^  +  .... 

Show  by  Ex.  4,  p.  516,  that  the  value  of  f  in  the  two  cases  is 

en  Vxzdn  \'\z 


^{z)  =  -  e^z  +  V\E{^,  k)  +  VX 


sn  VX 


f  (z)  =  -  (m  +  ei)  z  +  2  \^  E (0,  /.•)  +  ^r^ "'  ^  "^  ,-    (2  dn2  ^^^z  -  l), 

sn  Vyuz  dn  \' iiz 

where  X  =  (\  —  c„,         fc^  =  (e,  —  e.-,)/(ej  —  e.,),         0  =  sin-i  sn  v'Xz, 

and        /i  =  "\  (^1  —  e.3)(ej  —  Cg),         A:'-  =  1  —  3  ej/4  /x,         ^  =  sin-i  sn  \  fxz. 

2.  In  case  the  tliree  roots  are  real  show  that  p  (2)  —  e,-  is  a  square. 

cn^'Xz  I — — VX  / — ,.- dn  VXz 


\  p 


/-cn^^z  / — -— va  / — ■ ,.— 

(z)  -  t'l  =  VX -=- ,       Vp  (z)  -  e..  = ^  ,       Vp  (z)  -  ^3  =  \  X 


sn  A  X  z  sn  V  X  z  sn  \'  X  z 

What  happens  in  case  there  is  onlj"  one  real  root  ? 

3.  Letjij(z  ;  r/.,,  f/3)  denote  the  function  p  corresponding  to  the  radical 


Compute  p{\:  1 .  0),  p  (\  ;  0,  ^),  p  (| ;  13,  (3).    Solve  p  (2  ;  1,  0)  =  2,  p  (2  ;  0,  |)  =  3, 
p{z;  13.  0)  =  10. 

4.  If  0  of  the  9  points  in  which  a  cubic  cuts  y-  =  4  j^  —  ^.^j  —  g.^  are  on  a  conic, 
the  (ilher  three  are  in  a  straight  line. 

5.  If  a  conic  has  contact  of  the  second  order  with  the  cubic  at  two  points,  the 
points  of  contact  lie  on  a  line  through  one  of  the  infiections. 

6.  How  many  of  the  points  at  which  a  conic  may  have  contact  of  the  5th  order 
with  the  culiic  are  real  '.'    Locate  the  points  at  least  roughly. 

7.  If  a  conic  cuts  the  cubic  in  four  fixed  and  two  variable  points,  the  line  join- 
ing the  latter  two  passes  through  a  fixed  point  of  the  cubic. 

8.  Consider  the  space  curve  x  =  sn ;,  // =  en  <,  2  =  dn  <.  Show  that  to  each 
point  of  the  rectangle  4  A'  hy  UK'  corresponds  one  point  of  the  curve  and  con- 
versely. Show  that  the  curve  is  the  intersection  of  the  cylinders  x-  -f-  (/-  =  1  and 
k-.c-  +  z-  =  \.  Show  that  a  plane  cuts  the  curve  in  4  points  and  deternune  the 
relation  l)et\veen  the  jiarameters  of  the  points. 

9.  IIoVn-  many  osculating  planes  may  be  drawn  to  the  curve  (^f  Ex.  8  from  any. 
point  on  it?    At  how  many  puims  may  a  plane  have  contact  of  the  3d  order  with 
the  curve  and  where  are  the  points  ? 

10.   In  case  the  roots  are  real  show  that  i;{z)  has  the  form 

^1              ~      /     ~  \                       /~  ^       E^c^ 
f(r)z=    "2+  A  \Z,(\  Xz),         77i  =  \X£ ^. 

'^1  VX 


ELLIPTIC   FUXCTIOXS  52". 

Hence  log  a  (z)  =   f  ^(z)dz  =  ~'^  z^  +  log  n(-\/\z)  +  G 

'J  2  oij^ 

or  <r(2)=  Ce-"i'   ir(x\z). 

11.   By  general  methods  like  those  of  §  190  prove  that 

^  ^     [f(2  +  «)_f(._,,)_2i-(a)], 


P  (2)  -  P  (")  P'i'l) 

J  r  dz  1      ,       a-(z  +  (()        >2(-(a) 

and  I    = loi;  — ^— — -  +  2  ~^^-—!- . 

J    p{z)-p{a)  p'(a)      '    ff(z-n)         2)'{a) 

12.  Let  the  functions  ff  be  definetl  by  the.se  relations  : 

0  iz)  =  11  m .       ,,(,)  =  n(^ .       04Z)  =  e  (^) ,       ff,(z)  =  e,(^ 

with  fj  =  e  '"i  .  Show  that  the  ^-series  converge  if  Wj  is  real  ami  w.,  is  pure  imagi- 
nary or  complex  with  its  imaginary  part  positive.  Show  more  ueneraliy  that  the 
series  converge  if  the  angle  from  w,  to  oj,  is  positive  ami  less  than  180^. 

13.  Let  aiz)  =  e-4  '"  ^^~\  ,         a„(z)  =  c^^\  "'  ^"^'^ . 

^-(0)  '  ^.(0) 

Prove  (T(,r  -f-  2i<;j)  =  —  e"''i('^'"iV(4)  ami  similar  relations  for  (Xaiz). 

t  A       -r  -.  2  77,  OJ.T  n"i  TTL 

14.  Let  2  7?.,  =  — *— ,       or     tj^w.,  —  7),,Wj  =  — 

Wj  oij  "         '  2 

Prove  (r(2  +  2w.,)  =  —  t -''2*" '"2^(2)  and  similar  relations  for  0-^(2). 

15.  Sliow  that  cr(—  z)  =  —  (r(2)  and  develop  a (z)  as 

16.  With  the  determination  of  tj^  as  in  Ex.  15  prove  that 

^  log  „  ( .)  =  f  (z).  _  ^''-,  log  <j  iz)  =  -  r(2)  =  V  (2) 

((2      "  dz- 

liy  showing  that  p(.r)  as  here  define<l  is  doubly  periodic  witli  periods  2c<;p  2  Wj, 
with  a  pole  I/2-  of  the  second  order  at  2  =  0  and  witli  )io  constant  term  in  Its 
devehjpment.    State  why  this  identifies  j> (2)  with  the  function  of  t;he  text. 


CHAPTER    XX 

FUNCTIONS   OF   REAL  VARIABLES 

194.  Partial  differential  equations  of  physics.  In  the  solution  of 
pliysical  2)roblenis  partial  differential  equations  of  higher  order,  partic- 
ularly the  second,  frequently  arise.  "With  very  few  exceptions  these 
equations  are  linear,  and  if  they  are  solved  at  all,  are  solved  by  assum- 
ing the  solution  as  a  product  of  functions  each  of  -which  contains  only 
one  of  the  variables.  The  determination  of  such  a  solution  offers  only 
a  particular  solution  of  the  problem,  but  the  combination  of  different 
particular  solutions  often  suffices  to  give  a  suitably  general  solution. 
For  instance 

i?^;=o       (1) 

is  Laplace's  equation  in  rectangular  and  polar  coordinates.  For  a  solu- 
tion in  rectangular  coordinates  the  assum^jtion  1'=  X  {'■r')  VQ/)  would  l)e 
made,  and  the  assumption  F  =  7.' (/•)ci>((^j  for  a  solution  in  polar  coor- 
dinates. The  equations  would  then  become 

A'"       ]■"  r-Jl"         R'      ^" 

—  -f  —  =  0     or     +  /.  —  +  —  =  0.  (2) 

A'         Y  R  R        ^  ^  ^ 

Now  each  equation  as  written  is  a  sum  of  functions  of  a  single  variable. 
But  a  function  of  ,/■  cannot  equal  a  function  of  //  and  a  function  of  /■ 
cannot  equal  a  function  of  ^  unless  the  functions  are  constant  and  have 
the  same  value.    Hence 


c-V      c^V 

c-V      Ic] 

-7r-r  +  ^-r  = 

=  0 

or 

-^+-  — 

cx-        Clf 

Cj'-         /'    C/' 

X" 

4)  ~      '"  ' 

y" 

or 

;■-/.'"         R' 
R     +''a> 

(2') 
-f  vi". 

These  are  ordinary  ecpiations  of  the  second  order  and  may  be  solved 
as  such.    The  second  case  will  be  treated  in  detail. 
The  solution  (,'orresponding  to  any  value  of  ni  is 

<J)  =  f/^^^  cos  i/i<f>  +  />„,  sin  v)<f),  /'  =  .  I  ,„/•'"  +  J!,j-~ '" 

and  I'  =  7i'$  =  (-I,,,/-'"  +  R,j'~"")(",„  cos  ;//</>  +  /<„,  sin  /)><f>) 

524 


KEAL  VARIABLES  525 

or  T'  =  2  (-1  ,„'■"'  +  J'mi'~  "'){"m  COS  m<^  +  /v  sin  w<^).  (3) 

That  any  number  of  solutions  corresponding  to  different  values  of  m 
ma}'  be  added  together  to  give  another  solution  is  due  to  the  Unearitij 
of  the  given  equation  (§  96).  It  may  be  that  a  single  term  will  suffice 
as  a  solution  of  a  given  problem.  ]>ut  it  may  be  seen  in  general  that : 
A  solution  for  T'  inay  be  found  in  the  form  of  a  Fourier  series  which 
shall  give  V  any  assigned  values  on  a  unit  circle  and  either  be  conver- 
gent for  all  values  within  the  circle  or  be  convergent  for  all  values 
outside  the  circle.  In  fact  let  f(<i>}  be  the  values  of  V  on  the  unit  circle. 
Expand /(^)  into  its  Fourier  series 

f{^)  =  \  '■',1  +  "V  («■'„,  cos  iii<li  +  /»„,  sin  m<f). 
Then  T'  =  -^  a^  +  V  r'"  ('/,„  cos  ///^  +  /.,„  sin  w^)  (3') 

will  be  a  solution  of  the  equation  which  reduces  to  /(^)  on  the  circle 
and,  as  it  is  a  power  series  in  r,  converges  at  every  point  within  the 
circle.    In  like  manner  a  solution  convergent  outside  the  circle  is 

r  =  ],  r/„  +  V  /•-'"  ('/,„  cos  iii<^  -f  A,,,  sin  in(^).  (3") 

The  iiiiiiiite  series  for  V  have  l}een  called  solutions  of  Laplace's  equation.  As  a 
matter  of  fact  they  have  not  been  proved  to  be  solutions.  The  finite  sum  obtained 
by  taking  any  number  of  terms  of  the  series  would  surely  be  a  solution  ;  but  the 
limit  of  that  sum  when  the  series  becomes  infinite  is  not  thereby  ])rove(l  to  l)e  a  solu- 
tion even  if  the  series  is  convergent.  For  theoretical  purposes  it  would  be  necessary 
to  give  the  proof,  but  tlie  matter  will  be  passed  over  here  as  having-  a  neuiiuible 
bearing-  on  the  practical  solution  of  many  problems.  For  in  ]U'actice  the  values  of 
/((/))  on  tlie  circle  could  not  be  exactly  known  and  could  therefore  be  adecjuatcly 
represented  by  a  linite  and  in  general  not  very  large  number  of  terms  of  the  de- 
velopment of /(0),  and  these  terms  would  give  only  a  finite  series  for  the  desired 
function  1'. 

In  sonte  problems  it  is  better  to  keep  the  particular  solutions  se])a- 
rate,  discuss  each  possible  particular  solution,  and  then  imagine  them 
compounded  physically.  Thus  in  the  motion  of  a  drumhead,  the  most 
general  solution  obtainaV)le  is  not  so  instructive  as  the  particular  solution 
corresponding  to  particular  notes  ;  and  in  the  motion  of  the  surface  of 
the  ocean  it  is  jtreferable  to  discuss  individual  types  of  waves  and  com- 
])()und  them  according  to  the  law  of  superposition  of  snudl  vibrations 
([>.  22G).    For  example  if 

lf--_r-,v       c-x  1  7"_.V"       r"  _  vv^v' 


526  THEORY  OF   FUNCTIONS 

be  taken  as  the  equation  of  motion  of  a  rectangular  drumhead, 

_  fain  ax,  »- _  f^ii^ /^•''j  „,  _  fsin  c  Va-  -f-  ftH 

Loos  ax,  Lc.os  fix,  \cos  c  Va'-  +  fiH 


are  particular  solutions  which  may  be  combined  in  any  way  desired 
As  the  edges  of  the  drumhead  are  supposed  to  be  fixed  at  all  times, 

z  —  0     if     X  =  0,       ./•  =  ((,       y  ==  0,        y  =  ^>,        t  =  anything, 

where  the  dimensions  of  the  head  are  a  bv  h.  Then  the  solution 


rinr.r    .     viri/  \iir       n~ 

z  —  A'}'7'  =  sni sui  —~  cos  rir  \  —7  +  77  z' 

a  I)  >  c/-       Ir 


(-^) 


is  a  possible  type  of  vil)ration  satisfying  tlic  given  conditions  at  the 
])erimeter  of  the  lu'ad  foi'  any  integral  values  of  m ,  v.  The  solution  is 
])eriodic  in  f  and  rei)resents  a  ])articular  not,>  uliich  may  b'  omitted. 
A  sum  of  sucli  ex])i'(_'Ssions  multi[)li('(l  by  any  constants  would  also  be 
a  solution  and  would  represent  a  ])ossible  mode  of  motion,  luit  would 
not  be  periodic^  in  i  and  would  represent  no  note. 

195.    For  tlii't'c  dimensions  La})laci'"s  equation  l)ccomes 


cr\       cr 


sin'-^  rd)"       sin  9  cO\  c9 


0 


(5) 


in  ]iolar  cooi-dinates.    Substitute  V  —  /!  (r)(r){6)(i>((f)):  then 


A"  7r  ('"  ,lr  I  "^  (-)  sin  0  <I6  \  ''0  I  ^  *  sin^  ^  r/<^-  ~     ' 

Here   the    lirst   term   in\-olvcs   /•  alone  and    no   other   tciau    involves   r 
Hence  the  iirst  term  mnsv  be  a  constant,  sa\".  //(//  +  1).    Then 


/  /  .,  (in 


(),  +  1)/,'  =  0,  y.'  =  .!/•"  +  Jlr-"-\ 


Next  consider  the  last  term  after  multiplying  thr(.)Ugli  by  sin'-^.    It  ap 
})ears  that  4)~'(i)"  is  a  constant,  say, —  i/i'.     Ilenee 

<^"  =  ~  ///■<^.  <P  =  (•/„,  cos  i/i(f)  -f-  ''',„  sin  ni(f>. 

^loi'eovep  the  equation   foi'  0  now  reduces  to  the  sinqtle  form 


>/ 


(I  cos 


(^-<-'>^'^),/7 


/0 


("  +1) 


1- 


0  =  0. 


'I'iie  problem  is  now  separated  into  that  of  the  integration  of  three 
differential  equations  of  which  the  tirst  two  are  I'cadily  integrable.  The 
third  ei|iiation  is  a  generali/at  ion  of   Lcgendre"s  (Fxs.  13-17,  }).  252), 


EEAL   VARIABLES  527 

and  in  case  n,  m  are  positive  integers  the  solution  may  l:ie  expressed  in 
terms  of  polynomials  P„_  ,„  (cos  6)  in  cos  6.    Any  expression 

2  (-!„'•"  +  A,'-~"~')(«".  cos  7»<^  +  h,,  sin  ,),,f)  I\„,  (cos  $) 

n,  til 

is  therefore  a  solution  of  Laplace's  equation,  and  it  may  be  shown  that 
by  combining-  such  solutions  into  infinite  series,  a  solution  may  be 
obtained  wliich  takes  on  any  desired  values  on  the  unit  sphere  and 
converges  for  all  points  within  or  outside. 

Of  particular  simplicity  and  importance  is  the  case  in  which  1'  is  su}> 
posed  independent  of  (f>  so  that  )i/  =  0  and  the  equation  for  0  is  soluble 
in  terms  of  Legendre's  })olynomials  7'„(cos^)  if  n  is  integral.  As  the 
potential  V  of  any  distribution  of  matter  attracting  according  to  the  in- 
verse square  of  the  distance  satisfies  Laplace's  eqiiation  at  all  points 
exterior  to  the  mass  (§  201),  the  potential  of  any  mass  symmetric  with 
respect  to  revolution  about  the  polar  axis  ^  =  0  niay  be  expressed  if 
its  expression  for  points  on  the  axis  is  known.  For  instance,  the  poten- 
tial of  a  mass  M  distributed  along  a  circular  wire  of  radius  a  is 

r.l//-  1  r-        l-:^r'        1-3 -or" 

V «-  -f  /•■-  .V  (n       1  li^       1  ■  ;5  n''       1  .  .3  ■  5  <r 

l~i;"i>7  +  i>:4  7"2T4T^7  +  --''    ''>"^ 

at  a  point  distant  /•  from  the  center  of  the  wire  along  a  perpendicular 
to  the  plane  of  tlu;  wire.   Tlie  two  series 

I  M  /n  1  n^  1     .S  ,1'"  1  ..3.")  a' 

[tAI- ''o -  2  ?  ''--  +  Y'x  ,> ''."  ^TTTTi  7 ''«  +  ■■■ '      '•  >  "' 

are  then  precisely  of  the  form  2. 1  „/•"/•'„,  2. 1  „/•"""'/'„  admissible  for 
solutions  of  Laplace's  equation  and  reduce  to  the  known  value  of  T' 
along  the  axis  ^  =  0  since  P„(^)  —  ^-  I'l^*?}'  fti'^'6  the  values  of  F  at  all 
])()ints  of  space. 

To  this  point  the  method  of  combining  solutions  of  the  given  differ- 
ential equations  was  to  add  them  into  a  finite  or  infinite  series.  It  is 
also  possible  to  combine  tliem  by  integration  and  to  obtain  a  solution 
as  a  definite  integral  instead  of  as  an  inhnite  series.  It  should  be  noted 
in  this  case,  too,  that  a  limit  of  a  sum  lias  replaced  a  sum  and  that  it 
would  theoretically  be  necessary  to  demonstrate  that  the  limit  of  the 
sum  was  really  a  solution  of  the  given  equation.  It  will  lie  sufficient 
at  this  point  to  illustrate  the  luetliod  without  any  rigorous  attenq)t  to 


528  THEOIIY  OF  FUNCTIONS 

justify  it.  Consider  (2')  in  rectangular  coordinates.  The  solutions  for 
X,  Fare 

A'"  ,     F" 

where  F  may  be  expressed  in  terms  of  hyperbolic  functions.    Now 

I     c~ '""  \^a  (7?i)  cos  nix  +  h  (???.)  sin  mx^  chn 

(6) 
=  lim  2,  ''"'"''■'['■''  (?/?-,■)  cos  v)jX  +  f'('i'i)  Hin  ii/jx']  A»?,- 

is  the  limit  of  a  sum  of  terms  each  of  which  is  a  solution  of  the  given 
equation ;  for  a  (w?,)  and  b  (j)t,)  are  constants  for  any  given  value  ni  =  m,-, 
no  matter  Avhat  functions  a(m)  and  l)(m)  are  of  ?».  It  may  b(^  assumed 
that  V  is  a'  solution  of  the  given  equation.  Another  solution  could  be 
found  by  replacing  e~"'"  by  e'"". 

It  is  sometimes  })0ssible  to  determine  a  (i>i),  l>  (/»)  so  that  ]'  shall 
reduce  to  assigned  values  on  certain  lines.    In  fact  (p.  4()G) 

/(,r)  =-  I       j        /(A)  ('OS  VI (X  -  x)  (Ihlm.  (7) 

Hence  if  the  limits  for  m  be  0  and  co  and  if  the  (dioice 

a  (ill)  =  —   I        f(X)  cos  mX(/X,  h  (?ii)  =  —  |        fi^)  sin  mXdX 

is  taken  for  a  (m),  b(iii'),  the  expression  ((>)  for  V  becomes 

V  =  -  f      j        e-""'f(X)  cos  III  (X  -  x)  (iXdiii  (8) 

and  reduces  to /(,r)  when  // =  0.  Ibmce  a  solution  V  is  found  which 
takes  on  anv  assigned  values  ,/'(•'')  '^-^oi'g  ^'^'*'  ;r-axis.  Tliis  solution  clearly 
becomes  zei'o  when  y  be(H)mes  infinite.  AVhen  /(•'')  ^^  given  it  is  sonu'- 
times  possible  to  perform  one  or  more  of  the  integrations  and  thus 
simplify  the  ex})ression  for  V. 

For  instance  if 

/(.r)  =  T  when  j  >  0     and    /(.r)  =  0  wlien  x  <  0, 
tlie  intt'i^ral  from  —  co  t.o  0  drops  ont  and 

V=       f      f    e-'»'J  ■  1  •  c\mvL{\  — x)d\(li)i —-       f      f    c- '"■"  cos  ??i  (X  —  x)  d/nd\ 


1     r  '           v(l\                1  /tt               ,  •'■  \      -       1  ,  V 

/       ■  =  +  tan-i     U=  1 tan-i-. 

TT  J  0       //"  +  (X  —  X)-         TT  \2  I//  TT  X 


REAL  VARIABLES  529 

It  may  readily  be  shown  that  when  y  >  0  the  reversal  of  the  order  of  integration 
is  permissible ;  bnt  as  V  is  determined  completely,  it  is  simpler  to  substitute  the 
value  as  found  in  the  equation  and  see  that  F^^  +  1'^^  =  0,  and  to  check  the  fact 
that  V  reduces  to  f{x)  when  y  =  0.  It  may  perhaps  be  superfluous  to  state  that 
the  proved  correctness  of  an  answer  does  not  show  the  justification  of  the  steps  by 
which  that  answer  is  found;  but  on  the  other  hand  as  those  steps  were  taken 
solely  to  obtain  the  answer,  there  is  no  practical  need  of  justifying  them  if  the 
answer  is  clearly  right. 

EXERCISES 

1.  Find  the  indicated  particular  solutions  of  these  equations : 

(a)  c-  - —  =  - — -  ,         1"  =  "X  'l»it- '""'  («,„  cos  cmx  +  6„,  sin  anx), 
ct         ex'-  ^~-< 

(j3) = .         F  =  /    (^l„jC0sc??i^  +  7i,„si'i  cml)(amCosmx  +  6„iSin  )nx), 

c'^  ct'        dx-  -^ 

(y)  c-2^^  =  ^\^l\        ^-^.^  rsincax  ^.^jsinc/S^/  ^^,-(.=  +  ,2), 

^    '        ct        cx-^        cy-  icon  cax,  ^cos  cjSy, 

2.  Determine  the  solutions  of  Laplace's  equation  in  the  plane  that  have  V  =  1 
for  0  <  0  <  TT  and  F  =  —  1  for  it  <  4>  <2it  on  a  unit  circle. 

3.  If  Y  --\tt  —  4)]  on  the  unit  circle,  find  the  expansion  for  F. 

4.  Show  that  F=  Sff,„sinTO7rx/?  •  cos  cmirt/l  is  the  solution  of  Ex.  1  (/3)  which 
vanishes  at  x  =  0  and  x  =  I.  Determine  the  coefficients  a,„  so  that  for  t  =  0  the 
value  of  T"  shall  be  an  assigned  function /(x).  This  is  the  problem  of  the  violin 
string  started  from  any  assigned  configuration. 

5.  If  the  string  of  Ex.  4  is  .started  with  any  assigned  velocity  dV/ct  =f{x)  when 
t  =  0,  show  that  the  solution  is  2a„,  sin  nnrx/l  •  sin  cimrt/l  and  make  the  proper  deter- 
mination of  the  constants  am- 

6.  If  the  drumhead  is  started  with  the  shape  z  =.f{x,  y),  show  that 


■s:-\    ,  .    niTTX    .    mri/  ^       m-       n^ 

2=    >    .1,,,  ,,  sm sm — ^cosTTrfA/ 1 , 

f;^  a  b  \cfi       62 

A     r"^  r^'  viTTX    .     inry 

-4,„.„  =  --  f{x,y)sm sm——cbjdx. 

ab  J  0  J  0  a  b 

7.  In  hydrodynamics  it  is  shown  that  — ^  =  ' (/i6  -^)  is  the  differential  equa- 

ct-       b  cx  \      cxj 

tio]i  for  the  surface  of  the  sea  in  an  estuary  or  on  a  beach  of  breadth  b  and  depth 

li  measured  perpendicularly  to  the  x-axis  which  is  supposed  to  run  seaward.    Find 

(a)  y  =  ^4J"y(A.-x)cos  ni,       k-  =  11^ /gh,         {(3)  y  =  AJf^(2  \  lex)  cos  nt,       k  =  n-/gm, 

as  particular  solutions  of  tlie  equation  when  (cx)  the  depth  is  uniform  but  the 
breadth  is  proportional  to  the  distance  ovTt  to  sea,  and  when  (/3)  the  breadth  is  vini- 
form  but  the  depth  is  ntx.  Discuss  the  shape  of  the  waves  that  may  thus  stand  on 
the  surface  of  the  estiuiry  or  beach. 


530  THEORY   OF   FUNCTIONS 

8.  If  a  serifs  of  parallel  waves  on  an  ocean  of  constant  depth  h  is  cut  perpen- 
dicularly by  the  a"?/-plane  with  the  axes  horizontal  and  vertical  so  that  ?/  =  —  /i  is 
the  ocean  bed,  the  equations  for  the  velocity  potential  (p  are  known  to  be 

?l+?^=o,   p;*]    =0,   r§+„?*i  =0. 

Find  and  combine  particular  sc^lutions  to  show  that  (^  may  have  the  form 
(p  =  A  cosh  k{y  +  h)  cos  {kx  —  7t<),         ?i^  =  gk  taidi  kh. 

9.  Obtain  the  solutions  or  types  of  solutions  for  these  ecjuations. 

(fi)  ^~-  -\ 1-  —  ^ — -  +  V  =  0,  Am.  "V  (amcosm0-f-f'„,.si"'«0)t/m('')) 

(rf„,„,  cos  ?/i0 +  /)„,„,  sin  m0), 


dx'^         dy'-        cz'" 


■^V       JV      c'^V  ,  ,    1  c^V      c-V      f2]^      d-^v 

1-2 = ,  (e) -  = h h 

fr-  ct         dx^  c^    ct'^         cx'^         ci/^         cz^ 


10.  I-'ind  the  potential  of  a  homogeneous  circular  disk  as  (Kx.  22,  p.  68  ; 
Kx.  23,  p.  332) 

,,      2  3/  ri  a      1  •  1  a3  ^        1  ■  1  •  3  a^  i  .  ]  .  3  •  5  r;^  n 

V  = Po  + P. F,  +  ■  ■■],         r>a, 

a    12  r      2  •  4  r3     "      2  •  4  •  0  r^     *       2  •  4  •  6  •  8  r''     **  J  ' 

2  Jf  r,       r  „        1  '-2  ^        1  .  1  r*  1  •  1  •  3  r6  n 

= l^F-P,  -1- P„ P,  -h P, ,         r  <a, 

a    I        a     ^      2  «-^     -      2  ■  4  «4     ^  ^  2  •  4  •  (5  ««     'J 

where  the  negative  sign  before  P^  holds  for  6  <\Tr  and  the  positive  for  6  >  |-7r. 

11.  Find  the  potential  of  a  homogeneous  hemispherical  shell. 

12.  Find  the  potential  of  (a)  a  hf)mogeneous  hemisphere  at  all  points  outside 
the  hemisi)iiere,  and  (^)  a  homogeneous  circular  cylinder  at  all  external  pcjints. 

13.  Assume  —  cos-' is  the  potential  at  a  point  of  the  axis  of  a  conduct- 

2  n  x'^  -f  a^ 

ing  disk  of  radius  a  charged  with  (^  units  of  ehictricity.   Find  the  potential  anywhere. 

196.  Harmonic  functions ;  general  theorems.  A  function  wliieli 
satisfies  Laplace's  equation  I  "^.^  -f  !  'J^'^  =  0  or  I ",',.  -f  I  ",l,^  -f  I  ".'^  =  0,  Avlietlier 
in  the  plane  or  in  S})a('e,  is  cnllcd  a  Im rDinnin  fimcfinti.  It  is  assunu:-!! 
that  tlie  first  and  second  partial  derivatives  of  a  liiuMuouic  Function  are 
continuous  exce])t  at  S])ecihed  })oints  called  singailar  jjoints.  There  are 
many  similarities  between  harmonic  functions  in  the  ])lane  and  har- 
monic functions  in  space,  and  some  diiferiMict'S.  Tlie  fundamental  th"0- 
rem  is  that:  Jf  <(  finictloii  is  ha riiionic  and  ]i<ts  vo  si ii'inhi ritics  Vjinn. 
or  irifJii))  (t  si.iiijtlc  cIdsciI  ciiri-c  (nr  s/trfifcr^^  the  line  itdi'ijriil  of  i/s  iior- 
vial  (Jcrirnfiiw,  (iloiuj  tin',  ciirn'  (rcspcrfi rch/^  siirf'icc')  ranisltcs  \  <in<l  cm- 
V('/'seli/  if  (I  fiincfidii  !'(,/•,  //),  ar  r(.r,  _//,  rSV  has  continuous  Jirst  and  second 


REAL  VARIABLES  531 

partial  derivatives  and  the  line  integral  (or  sMrface  interjral)  along  everij 
closed  curve  (or  surface')  in  a  region  vanis/ws,  the  function  is  harmonic. 
For  by  Green's  Formula,  in  the  respective  cases  of  plane  and  space 
(Ex.  10,  p.  349), 


r  dv  ,       Ccv  ^      dv  ,       rr/c-v     c-]-\  ,  , 

1     -—ds^l     -^—  di/  —  -7—  dx  =111  ^r-r  +  — T7     dxdi/, 

rf'«=r"s-v.=///v-vr,,.,.,,,,fe. 


(9) 


Kow  if  the  function  is  harmonic,  the  right-hand  side  vanishes  and  so 
must  the  left;  and  conversely  if  the  left-hand  side  vanishes  for  all 
closed  curves  (or  surfaces),  the  right-hand  side  must  vanish  for  every 
region,  and  hence  the  integrand  must  vanish. 

If  in  particular  the  curve  or  surface  be  taken  as  a  circle  or  sphere  of 
radius  a  and  polar  coordinates  be  taken  at  the  center,  the  normal  de- 
rivative becomes  cV/cr  and  the  result  is 

/        ^  r/ci  =  0     or       r       I      ^.-  sin  OdOdch  =  0, 
Jo        ^'-  Jo      Jo      ''■ 

where  the  constant  a  or  ir  lias  been  discarded  from  the  element  of  arc 
((d<^  or  the  element  of  surface  <r  sin  ddddnfy.  If  these  equations  l)e  inte- 
grated Avith  respect  to  r  from  0  to  (f,  the  integrals  may  be  evaluated  by 
reversing  the  order  of  integration.    Thus 

0=   f\lr    r^'-Lj^^    C"   r"'±,frd^=  f'\\'.-  !;)./</>, 


and  j        l„./c^=  r,j       d^,     or      \\=V„  (10) 

Avhere  ]'„  is  the  value  of  1'  on  the  circle  of  radius  a  and  V^  is  the  value 
at  the  center  and  !'„  is  the  average  value  along  the  perimeter  of  the 
circle.  Similar  analysis  "would  hold  in  space.  The  result  states  the 
important  theorem:  Tlie  average  value  of  a  Jiarinonlc  function  over  a 
circle  (or  sjilierc)  is  e([ual  to  the  value  at  the  center. 

This  theorem  has  imm(»diat(.>  corollaries  of  imjxortance.  A  Jiarmonic 
function,  irltich  hus  no  singidaritics  irifliina  region  cunntd  become  maxi- 
itiuDi  or  minim uni  at  anij  poird  u-ithin  the  region.  I'or  if  the  function 
were  a  maximum  at  any  ])oint,  that  point  (H)idd  be  surroundi'd  by  a 
circli'  or  sphere  so  small  that  the  value  of  the  function  at  every  point 
of  the  contour  would  be  less  than  at  the  assumed  maximum  and  hence 
the  average  value  on  the  contour  coidd  not  be  the  value  at  the  center. 


532  THEORY  OF  FUNCTIONS 

A  ha n7i on ic  function  whic/i  Ikis  no  n'lngtilnrlfu's  irltJiln  a  region  and  is 
constant  on  the  boundari/  Is  constant  throughout  tlic  region.  Fur  the 
maximum  and  minimum  values  must  be  on  the  boundary,  and  if  these 
have  the  same  value,  the  function  must  have  that  same  value  through- 
out the  included  region.  Tu-o  lidrmonlc  functions  u-hldi  liare  identical 
values  upon  a  closed  contour  and  have  no  singula rltu's  u-ltltln,  are  ideii- 
tlral  throughout  the  Included  region.  For  their  difference  is  harmonic 
and  has  the  constant  value  0  on  the  boundary  and  hence  throughout 
the  region.  These  theorems  are  equally  true  if  the  region  is  allo-wed  to 
grow  until  it  is  infinite,  i)rovidc(l  the  values  -which  the  function  takes 
on  at  infinity  are  taken  into  consideration.  Thus,  if  two  hai'nionic 
functions  have  no  singularities  in  a  certain  infinite  region,  take  on  the 
same  values  at  all  points  of  the  l:)Oundary  of  the  region,  and  approach 
the  same  values  as  the  point  (.r,  y)  or  (,/■,  y,  z)  in  any  manner  recedes 
indefinitely  in  the  region,  the  two  functions  are  identical. 
If  Green's  Formula  be  applied  to  a  product  Ud  I  '/dn,  then 

r    dv  ,       r    dv  ,         dv  , 

I   i:  -—ds=  I   r  — -  dii  —  I    —-  dx 

Jo  '^''      Jo   '^•'-         '^y 

or  rrr/s.vr=  I  rv.vivr  +  I  vr.viv/r  (11) 

in  the  plane  or  in  space.  In  this  relation  let  ]'  be  harmonic  Avithout 
singularities  within  and  upon  the  contour,  and  let  /'  =  \'.  The  first  inte- 
gral on  the  right  vanishes  and  the  second  is  necessarily  })Ositive  unless 
the  relations  l'_^.  =  1',',  =  0  ov  \'',.  =  ',/ =  ^  ^  =  0,  which  is  equivalent 
to  V  r  =  0,  are  fulfilled  at  all  points  of  the  included  region.  Suppose 
further  that  the  normal  derivative  dV Jdn  is  zero  over  the  entire  bound- 
ary. The  integral  on  the  left  will  then  vanisli  and  that  on  the  right 
must  vanish.  Hence  )' contains  none  of  the  variables  and  is  constant. 
If  the  nariuiil  di'rlrntlcc  <if  a  fu nrthni  ha ciuonlc  and  dcmlil  (f  singula  f- 
Ifli's  at  all  jxtlnts  on  and  irlfJihi  a  glcmi  contour  ranlshrs  Identically 
upon  th,e  contour,  the  funrflon  Is  constmif.  As  a  corollary  :  If  tw>i 
functions  are  harmonic  and  devoid  of  singularities  upon  and  within  a 
given  contour,  and  if  tlieir  normal  derivatives  are  identically  equal 
upon  the  contour,  the  functions  differ  at  most  by  an  additive  constant. 
In  other  Avords,  a  harmonic  function  u-ltlmuf  singula rltlcs  not  only  Is 
dctcrnil ncd  hy  Its  ralncs  on  a  contour  luf  also  {c.fccpt  for  an  adiJltli'c 
constant)  hy  the  rallies  of'  Its  normal  dcrlratlcc  upon  a  contour. 


REAL  VARIABLES  533 

Laplace's  equation  arises  directly  upon  the  statement  of  some  problems  in 
physics  in  mathematical  form.  In  the  first  place  consider  the  flow  of  heat  or  of 
electricity  in  a  conducting  body.  The  physical  law  is  that  heat  flows  along  the 
direction  of  most  rapid  decrease  of  temperature  T,  and  that  the  amount  of  the  flow 
is  proportional  to  the  rate  of  decrease.  As  —  VT"  gives  the  direction  and  magni- 
tude of  the  most  rapid  decrease  of  temperature,  the  flow  of  heat  may  be  represented 
by  —  kV  T,  where  k  is  a  constant.  The  rate  of  flow  in  any  direction  is  the  compo- 
nent of  this  vector  in  that  direction.  The  rate  of  flow  across  any  boundary  is 
therefore  the  integral  along  the  boundary  of  the  normal  derivative  of  T.  Now  the 
flow  is  said  to  be  steady  if  there  is  no  increase  or  decrease  of  heat  within  aiij"  closed 

boundary,  that  is  p 

k  /  fZS-VT  =  0     or     T  is  harmonic. 

Hence  the  problem  of  the  distribution  of  tlie  temperature  in  a  body  supfjorting 
a  steady  flow  of  heat  is  the  jjroblem  of  integrating  Laplace's  equation.  In  like 
manner,  the  laws  of  the  flow  of  electricity  being  identical  with  those  for  the  flow 
of  heat  except  that  the  potential  V  replaces  the  temperature  T,  the  problem  of  the 
distribution  of  potential  in  a  body  supporting  a  steady  flow  of  electricity  will  also 
be  that  of  solving  Laplace's  equation. 

Another  problem  which  gives  rise  to  Laplace's  equation  is  that  of  the  irrotational 
motion  of  an  incompressible  fluid.  If  v  is  the  velocity  of  the  fluid,  the  motion  is 
called  irrotational  when  Vxv  =  0,  that  is,  when  the  line  integral  of  the  velocity 
about  any  closed  curve  is  zero.  In  this  case  the  negative  of  the  line  integral  from 
a  fixed  limit  to  a  variable  limit  defines  a  function  <f>  (x.  ij,  z)  called  the  velocity 
potential,  and  the  velocity  may  be  expressed  as  v  =— V<i>.  As  the  fluid  is  incom- 
pressible, the  flow  across  any  closed  boundary  is  necessarily  zero.    Hence 

CdS'V'^  =  0     or     fv.V^'Zr  =  0     or     V.V*  =  0, 

and  the  velocity  potential  <l>  is  a  harmonic  function.  Both  tliese  problems  may  be 
stated  without  vector  notation  by  carrying  out  the  ideas  involved  with  the  aid  of 
ordinary  coordinates.  Tlie  xiroblems  may  also  be  solved  for  the  plane  instead  of 
for  space  in  a  precisely  analogous  manner. 

197.  The  conception  of  the  flow  of  electricity  will  be  advantageous 
in  discussing  the  singularities  of  harmonic  functions  and  a  more  gen- 
eral conception  of  steady  flow.  Suppose 
an  electrode  is  set  down  on  a  sheet  of  zinc 
of  which  the  perimeter  is  grounded.  The 
equipotential  lines  and  the  lines  of  flow 
which  are  orthogonal  to  them  may  be 
sketched  in.  Electricity  passes  steadily 
from  the  electrode  to  the  rim  of  the  sheet 
and  off  to  the  ground.  Across  any  circuit 
wduch  does  not  surround  the  electrode  the 
flow  of  electricity  is  zero  as  the  flow  is  steady,  Imt  across  any  circuit 
surrounding  the  electrode  there  will  be  a  certain  definite  flow ;  the 
circuit  integral  of  the  normal  derivative  of  the  potential  1'  around  such 


584  THEOKY   OF  FTNCTIOXS 

a  circuit  is  not  zero.  This  may  be  compared  with  the  fact  that  the 
circuit  integral  of  a  function  of  a  complex  variable  is  not  necessarily 
zero  about  a  singularity,  although  it  is  zero  if  the  circuit  contains  no 
singularity.  Or  the  electrode  may  not  be  considered  as  corresponding 
to  a  singularity  but  to  a  portion  cut  out  from  the  sheet  so  that  the 
sheet  is  no  longer  simi)ly  connected,  and  the  comparison  would  then 
be  with  a  circuit  which  could  not  be  shrunk  to  nothing.  Concerning 
this  latter  interpretation  little  need  be  said ;  the  facts  are  readily  seen. 
It  is  the  former  conception  which  is  interesting. 

For  mathematical  purposes  the  electrode  will  be  idealized  by  assum- 
ing its  diameter  to  shrink  down  to  a  point.  It  is  physically  clear  that 
the  smaller  the  electrode,  the  higher  must  be  the  potential  at  the  elec- 
trode to  force  a  given  How  of  electricity  into  the  })late.  Indeed  it  may 
be  seen  that  T'  must  become  infinite  as  —  <_'  log  r,  where  r  is  the  distance 
from  the  point  electrode.  For  note  in  the  first  place  that  log  /•  is  a  solu- 
tion of  Laplace's  equation  in  the  ])lane  ;  and  let  U  =  V  -\-  C  log  /■  or 
r=  U  —  C  log  /•,  where  V  is  a  harmonic  function  which  reniains  finite 
at  the  electrode.  The  flow  across  any  small  circle  concentric  with  the 
electrode  is        ^-z-k  ^  y  /^ 2 -  -  , - 

—  j       ^—  j'>Jcf>  =  -  I        —  nl<p  +  2  7rr  =  2  ttC, 

and  is  finite.  Tlie  constant  '''  is  called  the  strength  of  the  source  situ- 
ated at  the  point  electrode.  A  simihir  discussion  for  space  would  show 
that  the  potential  in  the  neighljorhood  of  a  source  woidd  become  infinite 
as  C/r.  The  })articular  solutions  —  log  /■  and  1  '/•  of  Laplare's  e(|uation 
in  the  respective  cases  may  be  called  t\w  f/'>i'^"'>i''>tt"^  s'diit'tnns. 

The  physical  anah:)i;y  will  also  suggest  a  nu'thud  nf  dbtaiiiiiig  higher  singular- 
ities by  combining  fundamental  singularities.  F(ir  suppose  that  a  i)i)\verful  positive 
electrode  is  placed  near  an  ecjually  powerful  negative  electrode,  that  is.  suppose  a 
strong  source  and  a  strong  sink  near  together.  The  greater  part  of  tiie  tlow  will  be 
nearly  in  a  straight  line  from  the  source  to  the  sink,  but  some  part  of  it  will  spread 
out  over  the  .sheet.  The  value  of  T"  obtained  by  adding  together  the  two  values  for 
.source  and  sink  is 

r  =  -  1  C  log  {r-  4-  /-  -  2  H  cos  (p)  +  \  C  log  ( /•-  +  /-  +  2  rl  cos  4>) 
=  --Clog(^l-  -^-cos^  +  -,^  +  ^Clog^l  +  -cos,^  +  ^ 

2  IC  ,  .   ,  .V 

= cos  0  -f  hiuher  powers  =  — cos^  +  •  •  •• 

r  r 

Thus  if  the  strength  (_'  be  allowed  to  become  infinite  as  the  distance  2/  becomes 
ziTo.  and  if  M  denote  the   liuiir   of  the  ].roduct   •IlC.   the  linutiii-    form  of  b  is 
'  cos^  and  is  itself  a  soli u ion  of  the  ecjuation.  beconung  inliuiti-  more  strongly 
■■.    In  space  the  eo|-rrspoiidinu-  solution  would  be  3//'-- cos  (p. 


KEAL  VARIABLES  535 

It  was  seen  that  a  harmonic  function  \vhi(.-h  had  no  singularities  on  or 
within  a  given  contour  was  determined  by  its  values  on  the  contour  and 
determined  except  for  an  additive  constant  by  the  values  of  its  normal 
derivative  upon  the  contour.  If  now  there  be  actually  within  the  contour 
certain  singularities  at  which  the  function  becomes  infinite  as  certain 
particular  solutions  I'^,  l'^,  •  •  • ,  the  function  t'  =  T'  —  T^  —  I '.,—  •••  is  har- 
monic without  singularities  and  may  be  determined  as  before.  Moreover, 
the  values  of  I'^  1'.,,  •  •  •  or  their  normal  derivatives  may  be  considered  as 
known  upon  the  contour  inasmuch  as  these  are  definite  particular  solu- 
tions. Hence  it  appears,  as  before,  that  the,  harmonic  fu  net  ion  V  is  deter- 
mined hy  its  rallies  on  tJie  hoiindanj  of  the  region  or  {except  for  an  additire 
constant)  hy  tlie  values  of  its  normal  derivative  on  tlie  boundary,  provided 
the  singularities  are  specified  in pjosition  and  their  mode  ofbecom  ing  infin- 
ite is  given  m  earJi  case  as  some  particular  solution  of  La  pi  act;' s  equation. 

Consider  again  the  conducting  sheet  with  its  perimeter  grounded  and 
with  a  single  electrode  of  strength  unity  at  some  intericn-  ])oint  of  the 
sheet.  The  })otential  thus  set  up  has  the  properties  that :  1°  the  poten- 
tial is  zero  along  the  perimeter  because  the  perimeter  is  grounded  ;  2°  at 
the  position  P  of  the  electrode  the  })otential  becomes  infinite  as  —  log  ?■; 
and  3°  at  any  other  ])oint  of  the  sheet  the  potential  is  regular  and  sat- 
isfies Laplace's  equation.  Tliis  particular  distribution  of  potential  is 
denoted  b}'  G{P)  and  is  called  the  Green  Function  of  the  sheet  relative 
to  I\  In  space  the  Green  Function  of  a  region  would  still  satisfy  1°  and 
3°,  l)ut  in  2°  the  fundamental  solution  —  log  r  would  have  to  l)e  replaced 
Ijy  the  corresponding  fundamental  solution  1/r.  It  should  be  noted 
that  the  Green  Function  is  really  a  fun(,'tion 

G(P)  =  a(o^  b\  ./■,  y)     or      G{1')  =  G{a,  b,  r ;  x,  y,  z) 

of  four  or  six  variables  if  the  position  P(a,  h)  or  P  (<',  b,  c)  of  the  elec- 
trode is  considered  as  variable.  The  function  is  considered  as  known 
only  Avhen  it  is  known  for  any  position  of  J\ 

If  now  the  svmmetrical  form  of  Green's  Formula 


-ff(,,^r  -  .■s„yu,,  +£{„  ^  - . I),;,  =  0.        (12) 

where  A  denotes  the  sum  of  the  second  derivatives,  Ije  a^tplied  to  the 
entire  sheet  with  the  exception  of  a  small  circle  concentric  with  7-"  and 
if  the  choice  u  —  G  and  r  =  F  be  made,  then  as  G  and  ]'  are  harmonic 
the  double  integral  drojis  out  and 


r        da  ,        f-"    dv  r-^    dG 


(13) 


536  THEORY  OF  FUNCTIONS 

Now  let  the  radius  /■  of  the  small  circle  approach  0.  Under  the  assump- 
tion that  I'  is  devoid  of  singularities  and  that  G  becomes  infinite  as 
—  log  r,  the  middle  integral  approaches  0  because  its  integrand  does, 
and  the  final  integral  approaches  2  7rr(P).    Hence 

This  formula  expresses  the  values  of  T'  at  any  interior  point  of  the  sheet 
in  terms  of  the  values  of  V  upon  the  contour  and  of  the  normal  deri\'a- 
tive  of  G  along  the  contour.  It  appears,  therefore,  that  tJie  deter nibiatlon 
of  til e^  value  of  a  ho nnonlc  function  devoid  of  singularities  tvithin  and 
iipon  a  contour  tiunj  he  made  in  terms  of  the  values  on  the  contour  jj^'o- 
vided  the  Green  Function  of  the  region  is  known.  Hence  the  particular 
importance  of  the  problem  of  determining  the  Green  Function  for  a 
given  region.    This  theorem  is  analogous  to  Cauchy's  Integral  (§  126). 

EXERCISES 

1.  Show  that  any  linear  function  ax  +  by  +  cz  +  d  =  0  is  harmonic.  Find  the 
coiulitions  that  a  (quadratic  function  be  harmonic. 

2.  Show  that  the  real  and  imaginary  parts  of  any  function  of  a  complex  vari- 
able are  each  harmonic  functions  of  {x,  y). 

3.  Why  is  the  sum  or  difference  of  any  two  harmonic  functions  multiplied  by 
any  constants  itself  harmonic  ?    Is  the  power  of  a  harmonic  function  harmonic  '? 

4.  Show  that  the  product  J'l'  of  two  harmonic  functions  is  harmonic  when 
and  only  when  U^V^  +  ^"n^'n  —  ^  ^'''  V6'.V1"  =  0.  In  this  case  the  two  functions 
are  called  conjugate  or  orthogonal.  What  is  the  significance  of  this  condition 
geometrically  ? 

5.  Prove  the  average  value  theorem  for  space  as  for  the  plane. 

6.  Shiiw  for  the  jjlane  that  if  V  is  harmonic,  then 


f    =        -7—  '?N  =        ^^r~  dy  -  -z~ 
J     d)i  J     (X  cy 


dx 


is  independent  of  the  path  and  is  the  cunjugate  or  orthogonal  function  to  T',  and 
that  U  is  devoid  of  singularities  over  any  region  over  which  )"  is  devoid  of  them. 
Show  that,  1'  -f-  ii'  is  a  function  of  z  =  x  +  iy. 

7.  State  the  problems  of  the  steady  flow  of  heat  or  electricity  in  terms  of  ordi- 
nary coordinates  f(jr  the  case  of  the  plane. 

8.  Discuss  for  space  the  problem  of  the  source,  showing  that  C/r  gives  a  fiiute 
fl(jw  AttC.  where  C  is  called  the  strength  of  the  source.  Note  the  presence  of  the 
factor  4  77  in  the  place  of  2  tt  as  found  in  tW(j  dimensions. 

9.  Derive  the  solution  Mi—-  cos  (p  for  the  source-sink  combination  in  space. 


REAL   VARIABLES  537 

10.  Discuss  the  problem  of  the  small  magnet  or  the  electric  doublet  iu  view  of 
Ex.  9.  Note  that  as  the  attraction  is  inversely  as  the  square  of  the  distance,  the 
potential  of  the  force  satisfies  Laplace's  equation  in  space. 

11.  Let  equal  infinite  sources  and  sinks  be  located  alternately  at  the  vertices 
of  an  infinitesimal  square.  Find  the  corresponding  particular  solution  (a)  in  the 
case  of  the  plane,  and  (^)  in  the  case  of  space.  What  combination  of  magnets  does 
this  represent  if  the  point  of  view  of  Ex.  10  be  taken,  and  for  what  purpose  is  the 
combination  used  ? 

12.  Express  V{P)  in  terms  of  G{P)  and  the  boundary  values  of  T  in  space. 

13.  If  an  analytic  function  has  no  singularities  within  or  on  a  contour,  Caucliy's 
Integral  gives  the  value  at  any  interior  point.  If  tliere  are  within  the  contour  cer- 
tain poles,  what  must  be  known  in  addition  to  the  boundary  values  to  determine 
the  function  ?    Compare  with  the  analogous  tlieorem  for  harmonic  functions. 

14.  Why  were  the  .solutions  in  §  104  as  series  tlie  only  possible  .solutions 
provided  they  were  really  solutions?  Is  there  any  ditficulty  in  making  the  same 
infei'ence  relative  to  the  problem  of  the  potential  of  a  circular  wire  in  §  195  ? 

15.  Let  G{P)  and  G(Q)  be  the  Green  Functions  for  the  same  sheet  but  relative 
to  two  different  points  P  and  Q.  Apply  Green's  symmetric  theorem  to  the  .sheet 
from  which  two  small  circles  about  P  and  Q  have  been  removed,  making  the  choice 
u  =  G{P)  and  v  -  G{Q).  Hence  show  that  G  {P)  at  ^  '-^  t-ijual  to  (;{Q)  at  /'.  This 
may  be  written  as 

G{a,  b;  j,  y)  =  G{x.  y ;  a,  li)     or     6-' (a,  h.  c  ;  .c,  y,  z)  =  G  [x.  y.  z ;  a.  h^  c). 

16.  Te.st  these  functions  for  the  harmonic  property,  determine  tlie  conjugate 
functions  and  the  allied  functions  of  a  complex  variable: 

{ex)  xy,  (,a)  x-y  -  I  y^,  (7)  ^  log  {x-  +  y'^), 

(5)  e-^siuj,         (e)  sin  j  cosh  y.  (f)  tan-i(cot  x  tanh  y). 

198.  Harmonic  functions  ;  special  theorems.  For  the  purposes  of 
the  next  paragrii})lis  it  is  necessary  to  study  the  pro})erties  of  the  geo- 
metric transformation  known  as  Inrersuin.  Tlie  definition  of  inversion 
will  be  given  so  as  to  be  a})plieai)le  either  to  space  or  to  the  i)lane. 
The  transformation  which  replaces  each  point  P  by  a  point  P'  such 
that  OP  ■  OP'  =  Ir  where  o  is  a  given  fixed  point,  /.•  a  constant,  and  P' 
is  on  the  line  <>P,  is  called  inrersion  irlflt  flu-  n-nfcr  <)  and  tin;  rudlns  h. 
Xote  that  if  P  is  thus  carried  into  P\  then  /''  will  be  carried  into  P ; 
and  hence  if  any  geometrical  configuration  is  carried  into  another,  that 
other  will  be  carried  into  the  first.  Points  very  near  to  ()  are  carried 
off  to  a  great  distance;  for  the  point  O  itself  the  definition  l)reaks 
down  and  0  correspoiuls  to  no  point  of  space.  If  desired,  one  may  add 
to  space  a  fictitious  }ioint  called  the  point  at  infinity  and  niay  then  say 
that  the  center  O  of  the  inversion  corresponds  to  the  point  at  infinity 
(p.  481).  A  pair  of  points  /',  /''  which  go  over  iiitc;  each  other,  and  another 
pair  0,  0'  satisfy  the  equation  OP-  OP'  =  OQ-  Oil'. 


538 


THEORY  OF  FUNCTIONS 


A  curve  wliieh  cuts  tlie  line  oP  at  an  angle  t  is  carried  into  a 
curve  which  cuts  the  line  at  the  angle  t'  =  tt  —  t.  For  by  the  relation 
OP.  OP'  =  OQ.  OQ',  the  triangles  OPQ,  OQ'P'  are  similar  and 

Z  OPQ  =  Z  OQ'P'  =  TT  -  Z  0  —  Z  OP'Q'. 

Now  ii  Q  =  P  and  (/  =  P',  then  Z  0  =  0,  Z  OJ>Q  =  t,  Z  OP'Q'  =  t  and 
it  is  seen  that  t  =  tt  —  t'  or  t'  —  tt  —  t.    An  immediate  extension  of 
the  argument  will  show  that  the  magnitude 
of  the  angle  between  two  intersecting  curves  p____ — -;7f' 


will  be  unchanged  by  the  transformation;  tJu' 
transformation    is    therefore    confornial.     (In 
the  plane  where  it  is  possible  to  distinguish  \)etween  positive  and  neg- 
ative angles,  the  sign  of  the  angle  is  reversed  by  the  transformation.) 

If  polar  coordinates  relative  to  the  point  (>  be  introduced,  the  equations 
of  the  transformation  are  simply  ;■/■'  =  /■■-  Avith  the  understanding  that 
the  angle  <^  in  the  plane  or  the  angles  <^,  d  in  space  are  unchanged.  The 
locus  /■  =  /.■,  which  is  a  circle  in  the  plane  or  a  sphere  in  space,  becomes 
r'  =  1;  and  is  therefore  unchanged.  This  is  called  the  circle  or  the  sphere 
of  inversion.  Iielative  to  this  locus  a  simple  construction  for  a  pair  of 
inverse  points  /■•  and  /''  may  be  made  as  indicated  in  the  figure.   The  locus 


7-  +  /;-:=  2  V(r  +  A'-/- 


becomes     Ir  +  '''"  =  -  v  cr  +  /.■'/■'  cos  <^ 


and  is  therefore  unchanged  as  a  whole.  This  locus  represents  a  circle 
or  a  sphere  of  radius  a  orthogonal  to  the  circle  or  sphere  of  inversion. 
A  construction  may  now  l)e  made  for  hnding  an  inversion  which  car- 
ries a  given  circle  into  itself  and 
the  center  Z'  of  the  circle  into  any 
assigned  point  1^'  of  the  circle  ;  tlu' 
construction  holds  for  space  l)y  rc- 
voh'iug  tliehgurc  al>outtlie  \\\w(>l\ 

To  tiud  what  ligure  a  line  in  the  plane  or  a  })lane  in  space  becomes 
on  inversion,  let  the  polar  axis  <f>  =  0  or  ^  =  0  be  taken  perpendicular 
to  the  line  or  plane  as  the  casi^  i"'iy  1"'-    Then 
r  =  p  sec  (fi.  r'  sec  4>  =  /.'"/y/     or     /•  =  j/  sec  6,  r'  sec  0  =  A-'/y/ 

are  the  equations  of  the  line  or  })laiie  and  the  inverse  locus.  The  locus 
is  seen  to  be  a  circle  or  s})here  through  the  center  of  inversion.  This 
may  also  be  seen  directly  by  ap})lying  the  geometric  definition  of  in- 
versi(jn.  In  a  similar  manner,  or  analytically,  it  may  be  shown  that 
any  circle  in  the  })lane  or  any  sphere  in  s})ace  inverts  into  a  circle  or 
into  a  sphere,  unless  it  passes  through  the  center  of  inversion  and 
becomes  a  line  or  a  plane. 


KEAL  YAEIABLES  539 

If  d  be  the  distance  of  P  from  the  circle  or  sphere  of  inversion,  the  distance  of 
P  from  the  center  is  A;  —  d.  the  distance  of  P'  from  the  center  is  k'-/(k  —  d),  and 
from  the  circle  or  sphere  it  is  d'  =  dk/{k  —  d).  Now  if  the  radius  k  is  very  large 
in  comparison  with  d.  the  ratio  k/{k  —  d)  is  nearly  1  and  d'  is  nearly  equal  to  d. 
If  k  is  allowed  to  become  infinite  so  that  the  center  of  inversion  recedes  indefinitely 
and  the  circle  or  sphere  of  inversion  approaches  a  line  or  plane,  the  distance  d' 
approaches  d  as  a  limit.  As  the  transformation  which  replaces  each  point  by  a 
point  equidistant  from  a  given  line  or  plane  and  perpendicularly  opposite  to  the 
point  is  the  ordinary  inversion  or  reflection  in  the  line  or  plane  such  as  is  familiar 
in  optics,  it  appears  that  reflection  in  a  line  or  plane  may  be  regarded  as  the  limit- 
ing case  of  inversion  in  a  circle  or  sphere. 

The  importance  of  inversion  in  the  study  of  harmonic  functions  lies 
in  two  tlHH)rems  applicable  respectively  to  the  plane  and  to  space. 
First,  if  r  is  //"/•iDonic  orer  <niy  region  of  tlie  plune  and  If  tlud  region 
he  inrevti'd  in  "n;/  cin-lf,  the  function  T''(P')  —  V (^P)  forriu'iJ  Jn/  assigyi- 
ing  the  an  me  ralue  at  P'  in  the  new  region  as  the  funetion  had  at  the 
point  P  icliirh  inrevted  into  P'  is  also  liarmonie.  Second,  if  V  is  har- 
monic over  any  region  in  space,  and  if  that  region  he  inverted  in  a  sphere 
of  radius  k,  the  function  !''(/'')  =  ?:V(P')/r'  formed  hij  assigiiing  at  P' 
the  value  the  f /met ion  had  at  P  nndtijdied  hij  k  and  divided  hij  the  dis- 
tance OP'  =  /•'  of  P'  from  the  renter  of  inversion  is  also  harmonir.  The 
significance  of  these  theorems  lies  in  the  fact  that  if  one  distribution 
of  potential  is  known,  another  may  be  derived  from  it  by  inversion ; 
and  conversely  it  is  often  possible  to  determine  a  distriljution  of  ])oten- 
tial  by  inverting  an  unknown  case  into  one  that  is  knoAvn.  Tlu*  proof 
of  the  theorems  consists  merely  in  making  the  changes  of  variable 

V  =  k-/r'     or     /■'  =  //-//•,  cf>'  =  cj>,  6'  =  d 

in  the  polar  forms  of  Laplace's  equation  (Exs.  21,  22,  p.  112). 

The  method  of  using  inversion  to  determine  distribution  of  potential  in  electro- 
statics is  often  called  the  method  of  electric  imaijes.  As  a  charge  e  located  at  a 
point  exerts  on  other  point  charges  a  force  proportional  to  the  inverse  s(|uare  of 
the  distance,  the  potential  due  to  e  is  as  1/p.  where  p  is  the  distance  fmni  the 
charge  (with  the  proper  units  it  may  be  taken  as  e/p),  and  satisfles  Laplace's 
equation.  The  potential  due  to  any  number  of  point  charges  is  the  sum  of  the 
individual  iKitentiais  d\ie  to  the  charges.  Tims  far  the  theory  is  essentially  the 
same  as  if  tiie  charges  were  attracting  particles  of  matter.  In  electricity,  however, 
the  question  of  the  distribution  of  potential  is  furtlier  complicated  when  tliere  are 
in  the  neighl)orhood  of  the  charges  certain  conducting  surfaces.  Tor  1^  a  conduct- 
ing surface  in  an  electrostatic  field  must  everywhere  be  at  a  constant  potential  or 
there  would  be  a  component  force  along  the  surface  and  the  electricity  upon  it 
would  move,  and  2^  there  is  the  phenomenon  of  induced  electricity  whereby  a 
variable  surface  charge  is  induced  upon  tlie  conductor  by  other  cliarues  in  the 
neighborhood.  If  the  potential  ViP)  due  to  any  distriliati<iii  ef  diarges  be 
inverted  in  any  sphere,  the  new  potential   is  k]'{P)/r\    As  the  i>i)tential  V{P) 


540 


THEORY  OF  FUNCTIONS 


becomes  inlinite  as  e/p  at  the  point  cliarges  e,  the  potential  kV{P)/r'  will  become 
infinite  at  the  inverted  positions  of  the  charges.  As  the  ratio  ds' :  ds  of  the  in- 
verted and  original  elements  of  length  is  ?-'-/A:'^,. the  potential  kV{P)/r'  wiU  become 
infinite  as  k/r'  ■  c/p'  ■  r"^/k^,  that  is,  as  r'e/kp'.  Hence  it  appears  that  the  charge  e 
inverts  into  a  charge  e'  =  r'c/k  ;  the  charge  —  e'  is  called  the  electric  image  of  e. 
As  the  new  potential  is  ky{P)/r'  instead  of  V{P),  it  appears  that  an  eqnipoten- 
tial  surface  V  =  const,  will  not  invert  into  an  equipotential  surface  V'{P')  =  con.st. 
unless  V  =  0  or  r'  is  constant.  But  if  to  the  inverted  system  there  be  added  the 
charge  e  =  —  A;]'at  the  center  O  of  inversion,  the  inverted  equipotential  surface 
becomes  a  surface  of  zero  potential. 

With  these  preliminaries,  consider  tiie  question  of  the  distribution  of  potential 
due  to  an  external  charge  e  at  a  distance  r  from  the  center  of  a  conducting  spheri- 
cal surface  of  radius  k  which  has  been  grounded  so  as  to  be  maintained  at  zero 
potential.  If  the  system  be  inverted  with  respect  to  the  sphere  of  radius  k,  the 
potential  of  the  si^herical  surface  remains  zero  and  the  charge  e  goes  over  into  a 
charge  e'  =  r'c/k  at  the  inverse  point.  Now  if  p,  p'  are  the  distances  from  e,  e'  to 
the  sphere,  it  is  a  fact  of  elementary  geometry  that  p  :  p'  =  const.  =  ■K  :  ^•.  Hence 
the  potential 

)'         \p      kp' I  kpp' 


Y  = 


due  to  the  charge  e  and  to  its  image  —  e',  actually  vanishes  upon  the  sphere  ;  and 
as  it  is  harmonic  and  has  only  the  singularity  e/p  outside  the  sphere  (which  is  the 
same  as  the  singularity  due  to  e),  this  value  of  V  throughout  all  space  must  be 
precisely  the  value  due  to  the  charge  and  the  grounded  sphere.  The  distribution 
of  potential  in  the  given  .system  is  therefore  determined.  The  potential  outside 
tile  sphere  is  as  if  the  sphere  were  removed  and  the  two  charges  e,  —  e'  left  alone. 
By  (Jau.ss"s  Integral  (Ex.  8,  p.  348)  the  charge  within  iiny  region  may  be  evaluated 
by  a  surface  integral  around  the  region.  This  integral  over  a  surface  surrounding 
the  sphere  is  the  same  as  if  over  a  surface  shrunk  down  around  the  charge  —  c', 
and  hence  the  total  cliarge  induced  on  the  sphere  is  —  t'  —  —  r'c/k. 


199.   Inversion  will  tians'fonn  the  avera,f,^e  value  theorem 
1 


^\n  =  i7. 


V>/(fi     into      V'(P') 


-'^i 


T'V/^,  (14) 


a  form  applicable  to  determine  the  value  of  1'  at  any  point  of  a  eircle 

in  terms  of  the  value  upon  the  cireumferenee.    For  suppose  the  circle 

•with  center  at  /'  and  with  the  set 

of   radii   s[)aced   at   anj^des   dcf).   as 

im})lied  in  the  c()m})utation  of  the 

average  valu(%  he  inverted  up(Mi  an 

orthogonal  circle  so  chosen  that  7' 

shall  go  over  into  I''.    The   given 

circle  goes  over  into  itself  and  the  series  of  lines  goes  over  into  a  series 

of  cindes  through  /''  and  the  center  o  of  inversion.     (The  figures  are 

drawn  se})arately  instead  of  su{>erposed.j    From  the  confornud  jjropert}^ 


REAL  VARIABLES  541 

the  angles  between  the  circles  of  the  series  are  equal  to  the  angles  be- 
tween the  radii,  and  the  circles  cut  the  given  circle  orthogonally  just 
as  the  radii  did  Let  V  along  the  arcs  1',  2',  3',  ■  ■■  be  equal  to  V  along 
the  corresponding  arcs  1,  2,  3,  •  •  •  and  let  V(P)  =  V'(P')  as  required  by 
the  theorem  on  inversion  of  harmonic  functions.  Then  the  two  inte- 
grals are  equal  element  for  element  and  their  values  T'(P)  and  V'^P') 
are  equal.  Hence  the  desired  form  follows  from  the  given  form  as 
stated.  (It  may  be  observed  that  d(f>  and  dij/,  strictly  speaking,  have 
opposite  signs,  but  in  determining  the  average  value  V'(P'),  di(/  is  taken 
positively.)    The  derived  form  of  integral  may  be  written 

nn=Y^l    r#  =  ~j     rp,,  (14') 

as  a  line  integral  along  the  arc  of  the  circle.  If  P'  is  at  the  distance  r 
from  the  center,  and  if  a  be  the  radius,  the  center  of  inversion  O  is  at 
the  distance  a'^/r  from  the  center  of  the  circle,  and  the  value  of  k  is 
seen  to  be  k^  =  (a^  —  r^)«-//'".   Then,  if  Q  and  Q'  be  points  on  the  circle, 

^  ,         ^   OQ''       7\<r-2a'r''cosrt>'  +  a'r-"-)      ^^ 

ds   =  ds  —-r  =       7^, ^:^ — r, add). 

k-  {<r  —  v)  (C 

Now  d\p/ds'  may  be  obtained,  because  of  the  equality  of  d\p  and  d4>,  and 
ds'  may  be  written  as  ad(^'.    Hence 

^^       2'ir  X  «  -  2  ar  cos  <^'  +  ?•-'  ^ 

Finally  the  primes  may  be  dropped  from  V'  and  P',  the  position  of  P' 
may  be  expressed  in  terms  of  its  coordinates  (r,  ^),  and 

is  the  expression  of  T'  in  terms  of  its  boundary  values. 

The  integral  (15)  is  called  Polsson^s  Integral.  It  should  be  noted  par- 
ticularly that  the  form  of  Poisson's  Integral  first  obtained  by  inversion 
represents  the  average  value  of  1"  along  the  circumference,  provided  that 
average  be  computed  for  each  point  by  considering  the  values  along  the 
circumference  as  distributed  relative  to  the  angle  xp  as  independent  vari- 
able. That  T'  as  defined  by  the  integral  actually  approaches  the  value  on 
the  circumference  when  the  point  approaches  the  circumference  is  clear 
from  the  figure,  which  shows  that  all  except  an  infinitesimal  fraction  of 
the  orthogonal  circles  cut  the  circle  within  infinitesimal  limits  when  the 
point  is  infinitely  near  to  the  circumference.   Poisson's  Integral  may  be 


542  THEORY  OF  FUNCTIONS 

obtained  in  another  way.  For  if  /*  and  P'  are  now  two  inverse  y^oints 
relative  to  the  circle,  the  equation  of  the  circle  may  be  written  as 

p/p'  =  const.  =  r/a,     and      G  (P)  =  -  log  p  +  log  p'  +  log  (r/a)    (16) 

is  then  the  Green  Function  of  the  circular  sheet  because  it  vanishes  along 
the  circumference,  is  harmonic  owing  to  the  fact  that  the  logarithm  of  the 
distance  from  a  point  is  a  solution  of  Laplace's  equation,  and  becomes 
infinite  at  P  as  —  log  p.    Hence 


It  is  not  difficult  to  reduce  this  form  of  the  integi'al  to  (15). 

If  a  harmonic  function  is  defined  in  a  region  abutting  upon  a  segment 
of  a  straight  line  or  an  arc  of  a  circle,  and  if  the  function  vanishes  along 
the  segment  or  arc,  the  function  may  be  extended  across  the  segment 
or  arc  by  assigning  to  the  inverse  point  P'  the  value  V  (P')  =—  y{P), 
which  is  the  negative  of  the  value  at  P;  the  conjugate  function 


J     <^n  J     c.r     ^        c,/ 


(1') 


takes  on  the  same  values  at  /'  and  P'.  It  Avill  be  sufficient  to  prove 
this  theorem  in  the  case  of  the  straight  line  because,  by  the  theorem  on 
inversion,  the  arc  may  l)e  inverted  into  a  line  by  taking  the  center  of 
inversion  at  any  point  of  the  ar('  or  the  arc  produced.  As  the  La})lace 
operator  7)|  +  1)"^  is  independent  of  the  axes  (Ex.  25,  p.  112),  the  line 
may  be  taken  as  the  .r-axis  without  restricting  the  conclusion. 
Now  the  exteiidcil  funetinn  V [P')  satisfies  Laplace's  iMjuation  .since 


c.r  -  c>/  -  c.c-  cy- 

Therefore  V{P')  is  liarinonic.  V,y  tlic  detinitidii  ^'{P')  =  —■  l'(P)  and  the  assumption 
that  F  vanishes  alonu  the  segment  it  appears  tliat  the  function  1'  (in  tiie  two  sides 
of  the  Hne  pieces  on  tn  itself  in  a  continuous  manner,  and  it  remains  merely  to  sliow 
tliatit  pieces  on  to  itself  in  a  harmonic  manner,  that  is.  that  the  function  1' and 
its  extension  form  a  function  harmonic  at  points  of  the  line.  This  follows  from 
Poissoir.s  Integral  applied  to  a  circle  centered  on  the  line.    For  let 

JI{x,i/)-f'    ]'dip;     then     //(.r,  0)  =  0 

Jo 

because  Intakes  on  ecjual  and  ojiposite  values  on  the  u])p(u-  and  lower  semicircum- 
ferences.  Hence  JI  =  V{P)  =  V{P')  =  0  along  the  axis.  I5ut  //  =  !'(/')  along  the 
upper  arc  and  H  —  V{P')  along  the  lower  arc  because  Toisson's  Integral  takes  on 
the  boun(hiry  values  as  a  limit  when  tlu;  jioint  approacdies  the  boundary.  Now  as 
//  is  harmonic  and  agrees  with  ^'(P)  upon  the  whole  perimeter  of  the  upper  semi- 
circle it  nuist  be  identical  with   !'(/')  througlujut  that  semicircle.    Ju  like  manner 


KEAL  VARIABLES  543 

it  is  identical  with  V{P')  throughout  the  lower  semicircle.  As  the  functions  V(P) 
and  V{P')  are  identical  with  the  single  harmonic  function  H,  they  must  piece 
together  harmonically  across  the  axis.  The  theorem  is  thus  completely  proved. 
The  statement  about  the  conjugate  function  may  be  verified  by  taking  the  integral 
along  paths  symmetric  with  respect  to  the  axis. 

200.  If  a  function  tv  =z  f{z)  —  u  +  iv  of  a  complex  variable  becomes 
real  along  the  segment  of  a  line  or  the  arc  of  a  circle,  the  function  may 
be  extended  analytically  across  the  segment  or  arc  by  assigning  to  the 
inverse  point  P'  the  value  ic  —  7c  —  iv  conjugate  to  that  at  P.  This  is 
merely  a  corollary  of  the  preceding  theorem.  For  if  tu  be  real,  the 
harmonic  function  v  vanishes  on  the  line  and  may  be  assigned  equal 
and  opposite  values  on  the  opposite  sides  of  the  line ;  the  conjugate 
function  u  then  takes  on  equal  values  on  the  opposite  sides  of  the 
line.  The  case  of  the  circular  arc  would  again  follow  from  inversion 
as  before. 

The  method  employed  to  identify  functions  in  §§  185-187  w^as  to 
map  the  halves  of  the  ?r-plane,  or  rather  the  several  repetitions  of  these 
halves  which  were  required  to  complete  the  map  of  the  ^--surface,  on  a 
region  of  the  ,v-plane.  By  virtue  of  the  theorem  just  obtained  the  con- 
verse process  may  often  be  carried  out  and  the  function  w  —f(z) 
which  maps  a  given  region  of  the  .t-plane  upon  the  half  of  the  ?/>plane 
may  be  obtained.  The  method  will  apply  only  to  regions  of  the  «-plane 
which  are  bounded  by  rectilinear  segments  and  circular  arcs ;  for  it  is 
only  for  such  that  the  theorems  on  inversion  and  the  theorem  on  the 
extension  of  harmonic  functions  have  been  proved.  To  identify  the 
function  it  is  necessary  to  extend  the  given  region  of  the  s-plane  by 
inversions  across  its  boundaries  until  the  ^^'-surface  is  completed.  The 
method  is  not  satisfactory  if  the  successive  extensions  of  the  region  in 
the  5:-plane  result  in  overlapping. 

The  method  will  be  applied  to  determining  the  function  (a)  which 
maps  the  first  quadrant  of  the  unit  circle  in  the  s-plane  upon  the  upper 
half  of  the  «--plane,  and  (Ji)  which  maps  a  30°-60°-90°  triangle  upon  the 
upper  lialf  of  the  ?r-plane.    Sup- 
pose the  sector  ABC  mapped  on         t^^  ^' 
the  ?r-half-plane  so  that  the  perim- 


eter   ABC     corresponds     to    the      ^  t^////////^«  ^-^' 

real  axis  nJ,c.  When  the  perime- 
ter is  described  in  the  order  w^ritten  and  the  interior  is  on  the  left, 
the  real  axis  must,  by  the  principle  of  conformality,  be  described  in 
such  an  order  that  the  upper  half-plane  which  is  to  correspond  to  the 
interior  shall  also  lie  on  the  left.    The  points  a,  b,  c  correspond  to  points 


544  THEORY  OF  FUXCTIOXS 

.1,  B,  C.  At  these  points  the  correspondence  required  is  such  that  the 
conformality  must  break  down.  As  angles  are  doubled,  each  of  the 
points  A,  ]'>,  C  must  be  a  critical  point  of  the  first  order  for  w=f(z) 
and  a,  h,  c  must  l)e  branch  points.  To  map  the  triangle,  similar  con- 
siderations apply  except  that  whereas  '^''  is  a  critical  point  of  the  first 
order,  the  points  -1',  W  are  critical  of  orders  5,  2  respectively.  Each 
case  may  now  he  treated  separateh'  in  detail. 

Let  it  be  assumed  that  tlie  three  vertices  A,  B,  C  of  the  sector  po  into  the 
points*  10  —  0.  1.  X.  As  the  perimeter  of  the  sector  is  mapped  on  the  real  axis, 
the  function  vj—f{z)  takes  on  real  values  for  points  z  along  the  perimeter. 
Hence  if  the  sector  be  inverted  over  any  of  its  sides,  the  point  P'  wliich  corre- 
sponds to  P  may  be  triven  a  value  conjugate  to  lo  at 
P,  and  the  image  of  P'  in  the  H"-plane  is  symmetrical 
to  the  image  of  P  with  respect  to  the  real  axis.  The 
three  regions  V.  2\  3'  of  the  z-plane  correspond  to 
the  lower  half  of  the  wj-plane  ;  and  the  perimeters 
of  these  regions  correspond  also  to  the  real  axis. 
These  regions  may  now  be  inverted  across  their 
boundaries  and  give  rise  to  the  regions  2,  3,  4  which 
must  correspond  to  the  upper  half  of  the  i/'-plane. 
Finally  by  inversion  from  one  of  these  regions  the 

region  4'  may  be  ol)tained  as  corresponding  to  the  WyJ//^^ 

lower  half  of  the  )/'-plane.  In  this  manner  the  inver- 
sion has  been  carried  on  until  the  entire  r-plane  is  covered.  Moreover  there  is  no 
overlapping  of  the  regions  and  the  figure  may  be  inverted  in  any  of  its  lines  with- 
out producing  any  overlapping  ;  it  will  merely  invert  into  itself.  If  a  Riemann  sur- 
face were  to  be  constructed  over  the  ;/'-plane,  it  would  clearly  require  four  sheets. 
The  surface  could  be  connected  up  by  studying  the  correspondence  ;  but  this  is  not 
necessary.  Note  merely  that  the  function  ./'(-i)  becduies  infinite  at  ('  when  z  =  i 
bj'  hypothesis  and  at  C  when  z  =  —  i  by  inversion  :  and  at  no  other  point.  The 
values  ±  i  will  therefore  be  taken  as  poles  of /(2)  and  as  poles  of  the  second  order 
because  angles  are  thjubled.  Note  again  that  the  function /(^)  vanishes  at  ^1  when 
z  =  0  l)y  liypothesis  and  at  z  =  x  by  inversion.  These  will  lie  assiuned  to  be  zeros  of 
the  second  order  because  the  points  are  critical  points  at  which  angles  are  doubled. 
The  function 

w  =/(■)  =  cz^z-i)-^z  +  i)-^  =  (■z-{z-  + 1)-^ 

has  the  above  zeros  ami  poles  and  must  be  identical  with  the  desired  function  when 
the  constant  C  is  pniperly  chosen.  As  the  Cfirrespnudence  is  such  tliat/(l)  =  1  by 
hypothesis,  the  constant  C  is  4.  The  determination  of  the  function  is  complete  as 
given. 

Consider  next  the  case  of  the  triangle.  The  same  pmcess  of  inversion  and  re- 
peated inversion  may  be  f<jllowed,  and  never  results  in  overlapping  except  as  one 

*  It  may  be  observed  tliat  the  linear  trausfurniation  (y-  -\-  S)  "•'  —  cxn-  +  /3  (Ex.  15, 
p.  157)  has  tlu'ee  arbitrary  cinistiuits  tr:  f^:y:  5,  and  that  by  siirh  a  traiisforiiiatioii  any 
three  jtoints  of  the  ?'--i)lan('  may  be  carried  into  any  three  peiiits  of  the  7r'-jilane.  It  is 
tlierebire  a  jirnper  and  triviid  restriction  to  assume  tliat  0,  1,  x  are  the  points  of  the 
zc-plane  wliich  correspond  to  .1,  /.',  C. 


REAL  VAKIAI5LES 


545 


2iK' 


region  falls  into  absolute  coincidence  with  one  previously  obtained.   To  cover  the 

whole  z-plane  the  inversion  would  have  to  be  continued  indefinitely  ;  but  it  may 

be  observed  that  the  rectangle  inclosed  by  the  heavy  line 

is  repeated  indefinitely.  Hence  w  =  f(z)  is  a  doubly  periodic 

function  with  the  periods  2K,  2iK'  if  2 it,  2K'  be  the 

length  and  breadth  of  the  rectangle.   The  function  has  a 

pole  of  the  second  order  at  C  or  z  =  0  and  at  the  points, 

marked  with  circles,  into  which  the  origin  is  carried  by 

the  successive  inversions.   As  there  are  six  poles  of  the 

second  order,  the  function  is  of  order  twelve.   When  z  =  K 

at  A  or  z  =  iK'  at  A'  the  function  vanishes  and  each  of 

these  zeros  is  of  the  sixth  order  because  angles  are  increased 

G-fold.    Again  it  appears  that  the  function  is  of  order  12. 

It  is  very  simple  to  write  the  function  down  in  terms  of 

the  theta  functions  constructed  with  the  periods  2  A',  2  /A''. 


w=f{z)=C 


iif{z)eHz) 


H\z)e"^{z)ir^iz  -  a)ei{z  -  a)ir\z  -  lifQKz  -  13) 


For  this  function  is  really  doubly  periodic,  it  vanishes  to  the  sixth  order  at  A',  iK\ 
and  has  poles  of  the  second  order  at  tlie  i^oints 


0,      K  +  iK\      a  =  I  A  4-  1  iK',      cx  +  A'  +  iK',      p  =  2K  —'a, 

ir\z  +  a),  Q,{z  -13)  = 


As  p  =  2  K  —  a  the  reduction  II-{z  —  /3) 
be  made. 

w=f{z)=C 


/3  +  Jv  +  iK'. 
e^{z  +  a)  may 


//['(z)e«(2) 


ir^(z)e"^(z)ir^z  -  a)n\z  +  a)ef(z  -  a)ef(2  +  a) 


The  constant  C  may  be  deterinineil,  and  the  expression  for  f{z)  may  be  reduced 
further  by  means  of  identities;  it  might  be  expressed  in  terms  of  sn  (z,  k)  and 
en  (z,  k),  with  properly  chosen  fc,  or  in  terms  of  p{z)  and  p'(z).  For  the  purposes  of 
computations  that  might  be  involved  in  carrying  out  the  details  of  the  map,  it 
would  probably  be  better  to  leave  the  expres-sion  of  /(z)  in  terms  of  the  theta 
functions,  as  the  value  of  q  is  about  0.01. 


EXERCISES 

1.  Show  geometrically  that  a  plane  inverts  into  a  sphere  through  the  center  of 
inversion,  and  a  line  into  a  circle  through  the  center  of  inversion. 

2.  Show  geometrically  or  analytically  that  in  tlie  plane  a  circle  inverts  into  a 
circle  and  that  in  space  a  sphere  inverts  into  a  sphere. 

3.  Show  that  in  the  plane  angles  are  reversed  in  sign  by  inversion.    Show  that 
in  space  the  magnitude  of  an  angle  between  two  curves  is  unchanged. 

4.  If  (Z.s,  d.s',  dv  are  elements  of  arc,  surface,  and  volume,  show  that 


ds'  =  -  d.'i  =  —  ds,         dS'  =  —  dS  =  ~  (7.S, 


dv' 


h-  =  —  dv. 


N'lte  that  in  tlie  plane  an  area  and  its  inverted  area  are  of  opposite  sign,  and  that 
the  same  is  true  tif  vohuiies  in  space. 


546  THEORY  OF  FUXCTIOXS 

5.  Show  that  the  system  of  circles  through  any  point  and  its  inverse  with  respect 
to  a  given  circle  cut  that  circle  orthogonally.  Hence  show  that  if  two  points  are  in- 
verse with  respect  to  any  circle,  they  are  carried  into  points  inverse  with  respect  to 
the  inverted  position  of  the  circle  if  the  circle  be  inverted  in  any  manner.  In  par- 
ticular .show  that  if  a  circle  be  inverted  with  respect  to  an  orthogonal  circle,  its  cen- 
ter is  carried  into  the  point  which  is  inver.se  with  respect  to  the  center  of  inversion. 

6.  Obtain  Poisson's  Integral  (15)  from  the  form  (16').    Note  that 

clG  _  cos  (p.  n)      cos  {p'.  n)  _  a-  —  r^ 


r^  —  p'^  +  a-  —  2  ap  cos  (/a,  7i), 

dn  p  p'  u-p- 

7.  From  the  etjuation  p/p'  =  con.st.  =  r/a  of  the  .sphere  obtain 

1      a  1  ..^         In  y{a--  f^)  dS 


pro  4  7r«  J    r.,-2   i    ,.2 


P       >'  P  4  7r«  J    ^^^-2  ^  ,.2  _  2  ar  cos  (r,  «)]  t 

the  Green  Function  and  Poi.s.son's  Integral  for  the  sphere. 

8.  Obtain  roi.s.son"s  Integral  in  .space  by  the  method  of  inversion. 

9.  Find  the  potential  due  to  an  insulated  spherical  conductor  and  an  external 
charge  (by  placing  at  the  center  of  the  sphere  a  charge  equal  to  the  negative  of 
that  induced  on  the  grounded  .sphere). 

10.  If  two  spheres  intersect  at  right  angles,  and  charges  proportional  to  the 
diameters  are  placed  at  their  centers  with  an  opposite  charge  proportional  to  the 
diameter  of  the  common  circle  at  the  center  of  the  circle,  then  the  potential  over 
the  two  spheres  is  constant.  Hence  determine  the  effect  throughout  external  .space 
of  two  orthogonal  conducting  spheres  maintained  at  a  given  potential. 

11.  A  charge  is  placed  at  a  distance  h  from  an  infinite  conducting  plane. 
Determine  the  potential  on  the  supposition  that  the  plane  is  insulated  with  no 
charge  or  maintained  at  zero  potential. 

12.  Map  the  quadrantal  sector  on  the  upper  half-plane  .so  that  the  vertices 
C.  A.  B  correspond  to  1.  ck,  0. 

13.  Determine  the  constant  C  occurring  in  the  map  of  the  triangle  on  the  plane. 
Find  the  point  into  which  the  median  point  of  the  triangle  is  carried. 

14.  With  various  selections  of  correspondences  of  the  vertices  to  the  three  points 
0,  1,  00  of  the  ('.'-plane,  map  the  following  configurations  upon  the  upper  half-plane  : 

(a)  a  sector  of  (30^.         {j3)  an  i.sosceles  right  triangle, 

(>)   a  .sector  of  -i'r'.         (5)  an  equilateral  triangle. 

201.  The  potential  integrals.  If  p(.i\  //.  ;:)  is  a  function  defined  at 
different  points  of  a  region  of  S})ace,  the  integral 

evaluated  over  that  region  is  ealled  the  potential  of  p  at  tlie  point 
a.  rj.  ^).  The  significance  of  the  integi'al  may  be  seen  l>y  considering 
the  attraction  and  the  potential  energy  at  the  point  ($.  i-j.  t,)  due  to  a 


KEAL  VARIABLES 


547 


distribution  of  matter  of  density  p  (.r,  y,  z)  in  some  region  of  space. 
If  /i  be  a  mass  at  (|,  77,  ^)  and  in  a  mass  at  (,t,  y,  z),  the  component 
forces  exerted  by  m  upon  /x  are 


A'  =  c 


/i//^  .r 


l'  =  c 


/XHi  y  —  7; 


Z  =  c 


/z?»,  z  -  t, 


fim 


(19) 


and  7^  =  c  -—r  ?  T'  =  —  oli  —  +  C 

are  respectively  the  total  force  on  /x  and  the  potential  energy  of  the 
two  masses.  The  potential  energy  may  be  considered  as  the  work  done 
by  F  or  A',  Y,  Z  on  /x  in  bringing  the 
mass  fjL  from  a  fixed  point  to  the 
])oint  ($.  7).  I)  under  the  action  of  7/1 
at  (j:,  y,  z)  oy  it  may  be  regarded 
as  the  function  such  that  the  nega- 
tive of  the  derivatives  of  I'  by  ./•,  y,  z 
give  the  forces  A',  )',  Z,  or  in  vector 
notation  F=— Vl'.  Hence  if  the 
units  be  so  chosen  that  e  =  1,  and  if  y  ,/^ 
the  forces  and  potential  at  (L  -q.  I) 

be  measured  per  unit  mass  by  dividing  b}-  /x,  the  results  are  (after  dis- 
regarding the  arbitrar}-  constant  '") 


(.i^,r) 


H 


X  = 


ill  X  —  $ 


Y 


ill  y 


III   z 

Z  =  -,- 


I 


(19') 


Now  if  there  be  a  region  of  matter  of  density  p{.r,  y,  z),  the  forces  and 
potential  energy  at  (L  77.  0  measured  per  unit  mass  there  located  may 
be  obtained  bv  summation  or  intey-ration  and  are 


A 


:r-$)r/.rr/yi/z  ^,^_    rpifr 


(19") 


p(-':  y.  z){. 

lit  -  .rf  +  (-q 

It  therefore  appears  that  the  i)otential  ('  defined  Ijy  (IS)  is  the  negative 
of  the  potential  energy  T'  due  to  the  distribution  of  matter.*  Xote  fur- 
ther that  in  evaluating  the  integrals  to  determine  A',  }',  Z,  and  C  =  —  ]', 
the  variables  .r,  y,  z  with  respect  to  which  the  integrations  are  per- 
formed will  dro[)  out  on  substituting  the  limits  which  determine  tiie 
region,  and  will  tlierefore  leave  A',  }',  Z,  T  as  functions  of  the  param- 
eters 6.  -q.  t,  wliich  appear  in  the  integrand.    And  finally 

cl'  cU  dr 

-^^         Y=^-,         Z  =  ^-: 

C^  C-q  CQ 


X  = 


(20) 


*Iu  electric  ami  mairnetic  theory,  where  like  rfpel^  like,  the  potential  and  potential 
enersv  have  the  same  si^u. 


548  THEORY  OF  FUNCTIONS 

are  consequences  either  of  differentiating  Launder  the  sign  of  integration 
or  of  integrating  the  expressions  (19')  for  A',  1^,  Z  expressed  in  terms  of 
the  derivatives  of  U,  over  tlie  whole  region. 

Theokem.  The  potential  integral  U  satisfies  the  equations 

c-U      c-U      c^U      ^  c-U      c-r      c^U  ,  ,^,, 

known  respectively  as  Lajdace^s  and  Poissonh  Equations,  according  as 
the  point  (^,  -q,  t,)  lies  outside  or  within  the  body  of  density  p{x,  y,  z). 
In  case  (^,  r},  C)  lies  outside  the  body,  the  })roof  is  very  simple.  For 
the  second  derivatives  of  fJ  may  be  obtained  by  differentiating  with 
respect  to  $,  -q,  t,  under  the  sign  of  integration,  and  the  sum  of  the 
results  is  then  zero.  In  case  ($,  rj,  ^)  lies  within  the  body,  the  value 
for  r  vanishes  when  (^,  -q,  0  coincides  with  (j:,  y,  z)  during  the  integra- 
tion, and  hence  the  integrals  for  U,  X,  Y,  Z  become  infinite  integrals 
for  which  differentiation  under  the  sign  is  not  permissible  without  jus- 
tification. Suppose  therefore  that  a  small  sphere  of  radius  r  concentric 
Avith  (I,  7/,  t,)  be  cut  out  of  the  body,  and  the  contributions  F'  of  this 
sphere  and  F*  of  the  remainder  of  the  body  to  the  force  F  be  considered 
separately.  For  convenience  suppose  the  origin  moved  up  to  the  point 
(f,  t;,  0-  Then 

F  =  vr  =  F*  +  F'  =    r  pV  ^  dc  +  F'. 

Now  as  the  sphere  is  small  and  tlu^  density  p  is  supposed  continuous, 
the  attraction  7-''  of  the  sphere  at  any  point  of  its  surface  may  be  taken 
as  \  TTr^pJr,  the  quotient  of  the  mass  by  the  square  of  the  distance  to  the 
center,  where  p^  is  the  density  at  tlie  center.  The  force  F'  then  reduces 
t(j  —  ^^  TTp  T  in  magnitude  and  direction.    Hence 

v.F  =  v.vr  =  v.F*  +  v.F'  =    I    pV.V  -  dr  +  v.F 


■  =  J,v.vJ 


The  integral  vanishes  as  in  the  first  case,  and  V.F'  =  —  4  irp,.  Hence 
if  the  suffix  0  l>e  now  dropped,  V.vr  =  —  4  irp.  and  Poisson's  Ecjuation 
is  }»roved.    Gauss's  Integral  (p.  348)  affords  a  similar  proof. 

A  rigorous  treatment  of  the  potential  U  and  the  forces  A',  Y,  Z  and  tlieir  de- 
I'ivatives  requires  the  discussion  of  convergence  and  allied  topics.  A  detaile(l  treat- 
ment will  not  be  given,  hut  a  few  of  the  most  important  facts  may  be  pointed  out. 
Consider  the  ordinary  case  where  the  volume  density  p  n-mains  finite  and  the  body 
itself  does  not  extend  to  intinity.  The  integrand  p/r  bci'ouics  infinite  when  r  =  0. 
But  as  dv  is  an  infinitesimal  of  the  third  order  around  the  point  where  r  =  0,  the 
term  pdv/r  in  the  integral  l'  will  he  inhnitesimal.  may  l)e  disregarded,  and  the 
integral   U  converges.    In  like  manner  the  integrals  for  A',   Y.  Z  will  converge 


REAL  VARIABLES  549 

because  p{^  —  x)/f^,  etc..  become  infinite  at  r  =  0  to  only  the  second  order.  If 
cX/c%  were  obtained  by  differentiation  under  the  sign,  the  expressions  p//-^  and 
p(|  —  xf-/r>  would  become  infinite  to  the  third  order,  and  the  integrals 


C  ^dv  =   CCC  f^  r"  sin  d  drdcpdO,  etc., 


as  expressed  in  polar  coordinates  with  origin  at  r  =  0,  are  seen  to  diverge.    Hence 
the  derivatives  of  the  forces  and  the  second  derivatives  of  the  potential,  as  ob- 
tained by  differentiating  under  the  sign,  are  valueless. 
Consider  therefore  the  following  device  : 


1  _  _  c   1  cU 

r  ex  r  c^       J   '^  c^  r 


CU        /*     c    1  ,  r     c    I  , 

—  =  I  p dv  =  —  I  p dv, 

c|       J  ^  ci,  r  J  '^  ex  r 

+  p^'-,  -fp^^-dv^fl'Pd.-f^Pdv. 

ex  r  J      cx  r  J    r  cx  J  ex  r 


The  last  integral  may  be  transformed  into  a  surface  integral  so  that 
eU  _  r  1  ep  ,_    _  r  p  ^^^^_     ,.,  _  rrrlep 


ci 


f  i  1^  J,  _  f  e  cos  adS  =  fff  '-  '£  dxdydz  -  ff  P-  dydz.  (22) 


It  should  be  remembered,  however,  that  if  r  =  0  within  the  bodj',  the  transforma- 
tion can  only  be  made  after  cutting  out  the  singularity  r  =  0,  and  the  surface  inte- 
gral must  extend  over  the  surface  of  the  excised  region  as  well  as  over  the  surface 
of  the  body.  But  in  this  case,  as  dS  is  of  the  second  order  of  infinitesimals  while  r 
is  of  the  first  order,  the  integral  over  the  surface  of  the  excised  region  vanishes 
when  r  =  0  and  the  equation  is  valid  for  the  whole  region.    In  vectors 

VU=  f^dv-   f^dS.  (22') 

It  is  noteworthy  that  the  first  integral  gives  the  i)otential  of  Vp,  that  is,  the  inte- 
gral is  formed  for  Vp  just  as  (18)  was  from  p.  As  Vp  is  a  vector,  the  summation 
is  vector  addition.  It  is  further  noteworthy  tiiat  in  Vp  the  differentiation  is  with 
respect  to  x,  y,  z,  whereas  in  VU  it  is  with  respect  to  ^.  tj.  f.  Xow  differentiate 
(22)  under  the  sign.    (Distinguish  V  as  formed  for  ^,  rj.  f  and  x,  y,  z  by  V^  and  V^..) 

¥ = /  ff  7.  s '" '  f"  '•'" "  h  \ "'  "'■  ^'-^f  '■' = /  ^* '  •''"'""  -  /  "'* ,-  •"^- 

or  again  V^.V^  U=-   C  V^  ^  .V,p,7i-  +    C  pV,  -^-  .dS.  (2-3) 

This  result  is  valid  fur  the  whole  region.    Now  Uy  fJreen's  Furiunla  (Kx.  10,  p.  340) 

/ pv.r,  1  „.  +  /  V,  1  .v,p,„  =  / V,.  (pV.  1  j  „,  =  / ,v,  L,s  =  / p  1^  Us. 

Here  the  .small  region  about  ?•  =  0  must  again  be  excised  and  the  surface  integral 
must  extend  over  its  surface.  If  the  region  be  taken  as  a  sphere,  the  normal  dn, 
being  exterior  to  the  bod\',  is  directed  along  —  dr.    Thus  for  the  sphere 

f  p   '     -  lis  =   CC  p  —  r-  sin  edcpdd  =   CC  p  sin  $d(pdO  =  4  vp, 


550  THEOIIY   OF   FUNCTIONS 

where  o  is  the  average  of  p  upon  tlie  surface.    If  now  r  be  allowed  to  approach  0 
and  V«Vr-i  be  set  equal  to  zero,  Green's  Fornuila  reduces  to 


r  V^  -  'V.-  p'iv  =   {  P^x-  'dS  +  4  irp, 


where  the  volume  integrals  extend  over  the  whole  volume  and  the  surface  integral 
extends  like  that  of  (23)  over  the  surface  of  the  body  but  not  over  the  small  sphere. 
Hence  (23)  reduces  to  V.V  J7  =  —  4  irp. 

Throughout  this  discussion  it  has  been  assumed  that  p  and  its  derivatives  are 
continuous  throughout  the  body.  In  practice  it  frequently  happens  that  a  body 
consists  really  of  several,  say  two,  bodies  of  different  nature  (separated  by  a  bound- 
ing surface  6'j.,)  in  each  of  which  p  and  its  derivatives  are  continuous.  Let  the 
suffixes  1,  2  serve  to  distinguish  the  bodies.    Then  » 

The  discontinuity  in  p  along  a  surface  S^o  does  not  affect  a  triple  integral. 

vr  =  f^^dv,-  f  P^^as,.r2  +  f^'^^dv,-  f  P^ds,,,^. 

Here  the  first  surface  integral  extends  over  the  boundary  of  the  region  1  which 
includes  the  surface  iS'j.,  between  the  regions.  For  the  interface  <S'j„  the  direction 
of  tZS  is  from  1  into  2  in  the  first  case,  but  fmm  2  into  1  in  the  second.    Hence 


^•  =  /?'"-/^--/ 


^-~^-  dS, 


It  may  be  noted  that  the  first  and  second  surface  integrals  are  entirely  analogous 
because  the  first  may  be  regarded  as  extended  over  the  surface  separating  a  body 
of  density  p  from  one  of  density  0.  Now  V-VD"  maj-  be  found,  and  if  the  proper 
modifications  be  introduced  in  Green"s  Fornuda,  it  is  seen  that  V-VU  =  —  4wp 
still  holds  provided  the  point  lies  entirely  within  either  body.  Tlie  fact  that  p 
comes  from  the  average  value  p  ui)on  tlie  surface  of  an  intinitesinial  sphere  shows 
that  if  the  point  lies  on  the  interface  Nj.,  at  a  regular  point.  V«Vf'  =  —  4:Tr{lp^+  ]  p.,). 
The  application  of  (jreen's  Formula  in  its  synnnetric  form  (Kx.  10.  p.  340)  to 
the  two  functions  /—i  and  [',  and  the  calculation  of  the  integral  over  the  infini- 
tesimal .sphere  about  r  =  0,  gives 


/(.' 


_^,,       ^.„^1\,          r  (^  dV  ,.  '7   1\   ,  ,       ,     ^- 

V.vr—  L  V.V-)(ii- =    / T   \dS—47rli 

J    \r  dn  dn  r 

IU\        Idl 


ldU\ 

X'XU   ,         ^^    r\dn/T_       \dn,-.  ,  . 
-dS 


f'-^'"-X.f 


d   1 


-S/<''.-'-->;k7."'^--^'^' 


(2J) 


where  2  extends  over  all  the  surfaces  of  discontinuity,  including  the  boundar}-  of 
the  whole  body  where  the  density  changes  to  0.  Now  V«VL"=  —  -iirp  and  if  the 
definitions  l)e  ^iven  that 


\dnK      Xdnl.  ^ 


then 


KEAL  VARIABLES  551 


where  the  surface  integrals  extend  over  all  surfaces  of  discontinuity.  This  form  of 
U  appears  more  general  than  the  initial  form  (18),  and  indeed  it  is  more  general, 
for  it  takes  into  account  the  discontinuities  of  U  and  its  derivative,  which  cannot 
arise  when  p  is  an  ordinary  continuous  function  representing  a  volume  distribution 
of  matter.  The  two  surface  integrals  may  be  interpreted  as  due  to  surface  distribu- 
tions. For  suppo.se  that  along  some  surface  there  is  a  surface  den.sity  <r  of  matter. 
Then  the  first  surface  integral  re|)resents  the  potential  of  the  matter  in  the  .surface. 
Strictly  speaking,  a  surface  di.stribution  of  matter  with  o-  units  of  matter  per  unit 
surface  is  a  phy.sical  iiupos-sibility,  but  it  is  none  the  le.ss  a  convenient  mathemati- 
cal fiction  when  dealing  with  thin  sheets  of  matter  or  with  the  charge  of  electricity 
upon  a  conducting  surface.  The  .surface  di.stribution  may  be  regarded  as  a  limit- 
ing case  of  viilume  distriljution  where  p  becomes  infinite  and  the  volume  through- 
out wliich  it  is  ,'^pread  becomes  infiniteh'  thin.  In  fact  if  dn  he  the  thickne.ss  of 
the  sheet  of  matter  pdndS  =  a-dS.  The  second  surface  integral  may  likewise  be 
regarded  as  a  limit.  For  suppose  that  there  are  two  surfaces  infinitely  near  to- 
gether upitn  one  of  which  tiiere  is  a  surface  den.sitj-  —  <r.  and  upon  the  other  a  .surface 
density  cr.    The  potential  due  to  the  two  ecjual  .superimposed  elements  dS  is  the 

(T.dS,       ff.,dS..         ,    /I        ]\         ,,(/!,  ,     (Z   1  ,^_, 

-J — i  +  -^ — -  =  o-(/.S  ( )  =  ffdS dn  =  adn dS. 

i\  /■„  \r.2       )\/  dii  r  dn  r 

Hence  if  a-dn  —  r.  the  poteiUial  takes  the  form  rdi—'^/dndS.  Just  this  sort  of  dis- 
trilnuioii  of  magnetism  arises  in  the  case  of  a  magnetic  shell,  that  is.  a  surface 
covered  ou  one  side  with  positive  p(jles  and  on  the  other  with  negative  poles.  The 
three  integrals  in  (25)  are  known  resi)ectively  as  volume  jxitential.  surface  poten- 
tial, and  double  surface  potential. 

202.   The  jioteiitials  may  l)e  used  to  obtain  particular  integrals  of 
some  ditt'erential  equations.    In  the  Hrst  place  the  e(piation 

c-i'     c-r     c-r       ,,  ,     ,  -1    r  fdi' 

C.r  Cjl  CZ  \  IT  J  r 

as  its  solution,  when  the  integral  is  extended  over  the  i-egion  through- 
out which /'is  defined.  To  this  })articular  solution  for  l'  may  he  added 
tiny  solution  of  Laplace's  equation,  hut  the  particular  solution  is  fre- 
quently precisely  that  particular  solution  which  is  tlesired.  If  the 
functions  U  and  f  were  vector  functions  so  that  U  =  \l\  +  jr., -f  kt^^, 
and  f  =:  \f^  +  j/*,  +  k/lj,  the  results  would  be 

-7—;^^-;  +  -7r-;  =  i(:c,ii,z)     and     U  =  -, —        ; 

c.r-  CiJ-  CZ-  V    '  .^'      /  -AtT  J         V 

where  the  integration  denotes  vector  summation,  as  may  be  seen  by 
adding  the  results  for  V.Vr^  =  f^,  V.vr,,  =/„  V.VTg  =/;  after  multi- 
])lication  by  i,  j,  k.  If  it  is  desired  to  indictite  the  vectorial  nature  of 
U  and  f.  the  ]iotenti;d  U  may  be  called  a  vector  potential. 


552  THEORY  OF  FUNCTIONS 

In  evaluating  the  potential  and  the  forces  at  ($,  -q,  ^)  due  to  an  ele- 
ment dm  at  (dc,  y,  z),  it  has  been  assumed  that  the  action  depends  solely 
on  the  distan(;e  r.  Now  sup})ose  that  the  distribution  p  (x,  y,  z,  t)  is  a 
function  of  the  time  and  that  the  action  of  the  element  pdv  at  (x,  y,  z) 
does  not  make  its  effect  felt  instantly  at  ($,  rj,  ^)  but  is  propagated 
toward  ($,  -q,  t,)  fi'om  {x,  y,  z)  at  a  velocity  1/a  so  as  to  arrive  at  the  time 
{t  +  ar).  The  potential  and  the  forces  at  {^,  -q,  C)  ^^  calculated  by  (18) 
will  then  be  those  there  transpiring  at  the  time  t  +  ar  instead  of  at  the 
time  t.  To  obtain  the  effect  at  the  time  t  it  would  therefore  be  necessary 
to  calculate  the  jwtential  from  the  distribution  p  (x,  y,  z,  t  —  ar)  at  the 
time  t  —  ar.  The  potential 


•^^0'  +  (^-?/r  +  (C-.^)^ 


(26) 


where  for  brevity  the  variables  x,  y,  z  have  been  dropY)ed  in  the  second 
forni,  is  called  a  retarded  potential  as  the  time  has  been  set  back  from 
t  to  t  —  ar.   The  retarded  potential  satisfies  the  equation 

c'^U      c-U      d-U        ,d'U  .        .^       .    . 

according  as  (^,  rj,  C)  l'^^^  ivithin  or  outside  the  distribution  p.  There  is 
really  no  need  of  the  alternative  statements  because  if  ($,  t],  C)  is  out- 
side, p  vanishes.    Hence  a  solution  of  the  ecjuation 

cHl      c^U      d~U        ..c-U 

is  IJ  =  ^    [^^''^^'''-'"^10. 

4  TT    /  r 


The  proof  of  tlie  (■(nuvtion  (27)  is  relatively  simple.    For  in  vector  notation, 


v.^ii = v.v  r  p-^dv  +  v.v  r 


p(/)^,,.   ,  ^^   r  p{t-ar)-p{t) 


(Id 


i-n-p  +  V.V  C 


p{L-ar)-p{t) 


dv. 


Tiie  lirst  reduction  is  made  by  Toisson's  E(iuation.  The  second  expression  may 
be  evaluated  by  differentiation  under  the  si^n.  ¥i>v  it  should  l>e  remarked  that 
p{t  —  ar)—  p(t)  vanishes  when  r  =  0,  and  hence  the  order  of  the  infinite  in  the 
integrand  before  and  after  differentiation  is  less  by  unity  than  it  was  in  the  cor- 
respcmding  steps  of  §  201.  Then 

v,CP^^^^dD=  r/(-'^»^'^r:^!l^^'  +  r,(/-.r)-p(o]v,n.z., 

J  r  J     I  r  V  J 


REAL  VARIABLES  553 

+  {-a)p'Vtr.Vt-+  {-a)p'Vtr.Vt~  +  [p{t  -  ar)-  p{t)]Vt-V,~\dv. 

But  V^  =  -  V^     and     Vr  =  r/r  ami  V/- 1  =  —  r/r^     and     V.V/-~i  =  0. 

Hence  V^r.V^r  =  1,         V^r.V^r-i  =— r-2,         V^.Vjr  =  2r-i 

and      V.V  f  Pi^-<'>-)-P(Oa,  ^    r  '^a,  =    r^  '"P^'-'"\lv  =  a-^^'. 
^  r  J       r  J     r  ct-  H- 

It  wus  seen  (p.  345)  that  if  F  is  a  vector  function  with  no  curl,  that 
is,  if  VxF  =  0,  then  F'dr  is  an  exact  differential  (!<}> ;  and  F  may  be  ex- 
pressed as  the  gradient  of  (f>,  that  is,  as  F  =  \'<f>.  This  problena  may  also 
be  solved  by  potentials.    For  suj)pose 

- 1  r  v.F 

F  =  V<^,     then     V-F  =  V.V(/),  c^  =  - —        Jr.        (28) 

-iTT  J       r  ^     -^ 

It  appears  therefore  that  <^  may  be  expressed  as  a  potential.  This  solu- 
tion for  <^  is  less  general  than  the  former  because  it  depends  on  the 
fact  that  the  potential  integral  of  V«F  shall  converge.  jVIoreover  as 
the  value  of  <^  thus  found  is  only  a  particular  solution  of  V«F  =  V«V^, 
it  should  be  proved  that  for  this  <^  the  relation  F  =  V^  is  actually  sat- 
isfied. The  proof  will  be  given  below.  A  similar  metliod  may  now  be 
employed  to  show  that  if  F  is  a  vector  function  with  no  divergence, 
that  is,  if  V«F  =  0,  then  F  may  be  written  as  the  curl  of  a  vector 
function  G,  that  is,  as  F  =  VxG.    For  suppose 

F  =  VxQ,      then      V^F  =  VxVxQ  =  VV-G  -  V-VG. 

As  G  is  to  be  determined,  let  it  be  sup]wsed  that  V'G  =  0. 


Then  F  =  VxG     gives      G 


1  r  vxF 


Here  again  the  solution  is  valid  only  when  tlie  vector  potential  integral 
of  VxF  converges,  and  it  is  further  necessary  to  show  that  F  =  VxG. 
The  conditions  of  convergence  are,  however,  satisfied  for  the  functions 
that  usually  arise  in  physics. 

To  amplify  the  treatment  of  (28)  and  (29),  let  it  be  shown  that 

1         r  V'F  1  r  VxF 

V0  = V  I   ^^dv^F,         VxG  =  -  -  Vx   (    ^^Jv  =  F. 

By  use  of  (22)  it  is  possible  to  pass  the  differentiations  under  the  si.un  of  integra- 
tion and  apply  them  to  the  functions  V.F  and  VxF,  instead  of  to  1/r  as  would  be 
required  by  Leibniz's  Rule  (§  119).    Then 


1     r  ^^'F  ,  1      r  ^-F  ,o 

v<t>  =  -~-  \ jy  +       /  -  -  as. 

iir  J        r  -iir  'J      r 


554  THEORY  OF  FUXCTIOXS 

The  surface  integral  extends  over  the  surfaces  of  discontinuity  of  V^F,  over  a  large 
(infinite)  surface,  and  over  an  infinitesinial  sphere  surrounding  r  =  0.  It  will  be 
assumed  that  V.F  is  such  that  the  surface  integral  is  infinitesimal.  Now  as  VxF  =  0, 
VxVxF  =  0  and  VV.F  =  V.VF.  Hence  if  F  and  its  derivatives  are  continuous,  a 
reference  to  (24)  shows  that 


1    r  ^'"^F  , 

V.;&  = I dv  -  F. 

■iir  J        r 

„  ^       1     /'VxVxF,         1     /-VxF     ,„      -1    /-v.v; 
VxG  =  —   I dv I    xdS  = I 


In  like  manner 

V.VF 


dv 


Questions  of  continuitj-  and  tlie  significance  of  the  vanishing  of  the  neglected  sur- 
face integrals  will  not  be  further  examined.  The  elementarj'  facts  concerning 
potentials  are  necessary  knowledge  for  students  of  physics  (especially  electro- 
magnetism)  ;  the  detailed  discussion  of  the  subject,  whether  from  its  physical  or 
mathematical  side,  may  well  be  left  to  special  treatises. 

EXERCISES 

1.  Discuss  the  potential  U  and  its  derivative  VI7  for  the  case  of  a  uniform 
sphere,  both  at  external  and  internal  points,  and  upon  the  surface. 

2.  Discuss  the  second  derivatives  of  the  potential,  that  is,  the  derivatives  of  the 
forces,  at  a  surface  of  discontinuity  of  density. 

3.  If  a  distribution  of  matter  is  external  to  a  sphere,  tlie  average  value  of  the 
potential  on  the  .spherical  .surface  is  the  value  at  the  center  ;  if  it  is  internal,  the 
average  value  is  the  value  obtained  by  concentrating  all  the  mass  at  the  center. 

4.  What  density  of  distribution  is  indicated  by  the  potential  e-'''  ?  AVhat  den- 
sity of  distribution  gives  a  potential  proportional  to  itself  '.' 

5.  In  a  space  free  of  matter  the  determination  of  a  putenthil  wliicli  shall  take 
assigned  values  on  the  boundary  is  equivalent  to  the  problem  of  minimizing 


ifffm-m^mh--if 


vr.VTcZr. 


6.  F'lr  Laplace's  equation  in  the  plane  and  for  the  logarithmic  potential  —  log  r, 
develop  the  theory  of  potential  integrals  analogously  to  the  work  of  §  201  for 
Laplace's  equation  in  space  and  for  the  fundamental  solution  1/r. 


BOOK  LIST 

A  short  list  of  typical  hooks  with  V)rief  comments  is  given  to  aid  the 
student  of  this  text  in  selecting  material  for  collateral  reading  or  for 
more  advanced  study. 

1.  Some  standard  elementary  differential  and  integral  calculus. 

For  reference  the  book  with  whicli  tlie  student  is  familiar  is  probably  preferable. 
It  may  be  added  that  if  the  student  has  had  the  misfortune  to  take  his  calculus  under 
a  teacher  who  has  not  led  him  to  acquire  an  easy  formal  knowledge  of  the  subject, 
he  will  save  a  great  deal  of  time  in  the  long  run  if  he  makes  up  the  deficiency  soon 
and  thoroughly;  practice  on  the  exercises  in  Granville's  Calculus  (Ginn  and  Com- 
pany), or  Osborne's  Calculus  (Heath  &  Co.),  is  especially  recommended. 

2.  B.  ().  Peirce,  Table  of  Inter/ rah  (new  edition).  Ginn  and  Company. 

This  table  is  frequently  cited  in  the  text  and  is  well-nigh  indispensable  to  the 
student  for  constant  reference. 

3.  Jahxke-Emde,  FunktlonentafeJn  in  it  Fornieln  xind  Kurven. 
Teubner. 

A  very  useful  table  for  any  one  who  has  numerical  results  to  obtain  from  the 
analysis  of  advanced  calculus.  There  is  very  little  duplication  between  this  table 
and  the  previous  one. 

4.  Woods  and  Bailky,  Course  in  Mathematicsi.    (linu  and  Com])aiiy. 

5.  BvKin.Y,  Diffrrcntial  Cah'ulas  and  Integral  Calculns.  Ginn  and 
Company. 

6  ToDiiuxTEK,  Differential  Calculus  and  Integral  Calculus.  ^Mac- 
millan. 

7.  Williamson,  Differential  Calculus  and  Integral  Calculus.  Long- 
mans. 

These  are  standard  works  in  two  volumes  on  elementary  and  advanced  calculus. 
As  sources  for  additional  problems  and  for  comparison  with  the  methods  of  the 
text  they  will  prove  useful  for  reference. 

8.  C.  J.  DE  LA  YALLp:E-Porssix,  Cours  (V  analyse.    Gauthier-Villars. 

Then;  are  a  few  books  which  inspire  a  positive  affection  for  their  .style  and 
beauty  in  addition  to  re.spect  for  their  contents,  and  this  is  one  of  those  few. 
]My  Advanced  Calculus  is  necessarily  under  considerable  obligation  to  de  la  Vall^e- 
Poussin's  Cours  d'  analyse,  because  I  taught  the  subject  oi;t  of  that  book  for  several 
years  and  esteem  the  work  more  highly  than  any  of  its  compeers  in  any  language. 

555 


556  BOOK  LIST 

9.   GouKSAT,  Cours  (V  analyse.    Gauthier-Villars. 

10.  Gouksat-Hkdkkk,  Mathematical  Anali/sis.    Ginn  and  Company. 

The  latter  is  a  translation  of  the  first  of  the  two  volumes  of  the  former.  These, 
like  the  preceding  five  works,  will  be  useful  for  collateral  reading. 

11.  ])KKTKAND,  Cdlcul  dlfferentiel  and  Calcul  Integral. 

This  older  French  work  marks  in  a  certain  sense  the  acme  of  calculus  as  a 
means  of  obtaining  formal  and  numerical  results.  Methods  of  calculation  are  not 
now  so  prominent,  and  methods  of  the  theory  of  functions  are  coming  more  to  the 
fore.  Whether  this  tendency  lasts  or  does  not,  Bertrand's  Calculus  will  remain  an 
inspiration  to  all  who  consult  it. 

12.  FoKSYTH,  Treatise  on  Different  la  J  J'^'/i/atlons.   ]Macmillan. 

As  a  text  on  the  solution  of  differential  equations  Forsyth's  is  probably  tlic 
best.  It  may  be  used  for  work  complementary  and  supplementary  to  Chapters 
VIII-X  of  this  text. 

13.  PiEiM'OXT,  TJieorij  of  Funetloyis  of  Ileal  Varlahles.  Ginn  and 
Company. 

In  some  parts  very  advanced  and  diilicult,  but  in  otiicrs  quite  elementary  and 
readable,  this  work  on  rigorous  analysis  will  be  found  useful  in  connection  with 
Chapter  II  and  other  theoretical  portions  of  our  text. 

14.  GiBBs-WiLSox,  Vector  Anahjsls.    Seribners. 

Herein  will  be  found  a  detailed  and  connected  treatment  of  vector  methods 
mentioned  here  and  there  in  this  text  and  of  fundamental  importance  to  the 
mathematical  physicist. 

15.  ]).  ().  Pkikck,  Xi'irtonlan  Potential  Function.   Ginn  and  Comi)any. 

A  text  on  the  use  of  tlie  x>'>tential  in  a  wide  range  of  physical  problems.  Like 
the  following  two  works,  it  is  adapted,  and  practically  indispensable,  to  all  who 
study  higher  matliematies  for  the  use  they  may  make  of  it  in  practical  x>i'oblems. 

1().  ]>YKKLY,  Fourier  Series  and  Splicrlcal  Ilarmonlcs.  Ginn  and 
Company. 

of  international  repute,  this  book  presents  the  methods  of  analysis  employed 
in  tlie  solution  nf  the  differential  ecjuations  of  physics.  Like  the  foregoing,  it  gives 
an  extended  development  of  some  questions  briefly  treated  in  our  Chapter  XX. 

17.   AViiiTTAKKi;,  Moil  em  Annhjsls.    Cambridge  University  Press. 

This  is  probably  tlie  only  book  in  any  language  which  develops  and  applies  the 
methods  (if  the  tlieory  of  functions  for  the  purpose  of  deriving  ami  studying  the 
formal  properties  of  the  most  inqiortant  functions  t)tlier  than  elementary  which 
occur  in  analysis  directed  toward  the  needs  of  the  applied  mathematieian. 

IS.   ()sGO()i),  Lt'hrhurli  tier  Funh'flnni'ntheorlc.    Ten1)nt'r. 
Fiu-  the  pure  niatlifinatician  this  work,  written  with  a  grace  comparable  only 
to  that  of  de  la  Vallee-roussin"s  Calculus,  will  be  as  useful  as  it  is  charming. 


INDEX 


(The  muiibeis  refer  to  pages) 


a^,  a',  4,  45,  162 

AbeFs  theorem  on  uniformity,  438 

Absolute  convergence,  of  integrals,  357, 
369  ;  of  series,  422,  441 

Absolute  value,  of  complex  numbers, 
154;  of  reals,  35;  sum  of,  36 

Acceleration,  in  a  line,  13;  in  general, 
174;  problems  on.  186 

Addition,  of  complex  numbers,  154 ;  of 
operators,  151 ;  of  vectors,  154,  163 

Adjoint  equation,  240 

Algebra,  fundamental  theorem  of,  159, 
306,  482  ;  laws  of,  153 

Alternating  series,  39,  420,  452 

am  =  sin-i  sn,  507 

Ampere's  Law,  350 

Amplitude,  function,  507;  of  complex 
numbers,  154 ;  of  harmonic  motion, 
188 

Analytic  continuation,  444.  543 

Analytic  function,  304,  435.  .See  Func- 
tions of  a  complex  variable 

Angle,  as  a  line  integral,  297,  308  ;  at 
critical  points,  491 ;  between  curves, 
9 ;  in  space.  81 ;  of  a  complex  number, 
154;  solid,  347 

Angidar  velocity,  178.  346 

Approximate  formulas.  60,  77,  101,  383 

Approximations,  59, 195;  successive,  198. 
See  Computation 

Arc,  differential  of.  78.  80. 131 ;  of  ellipse, 
77,  514  :  of  hyperbola.  516.  See  Length 

Area,  8,  10.  25,  67,  77;  as  a  line  integral, 
288;  bv  double  integration.  324,  329; 
directed,  167;  element  of,  80, 131, 175, 
340,  342  :  general  idea,  311;  of  a  sur- 
face. 339 

Areal  velocity,  175 

Argument  of  a  complex  ntimber.  154 

Associative  law,  of  addition,  153, 163  ;  of 
multiplication,  150,   153 

Asymptotic  expansion,  390.  397,  456 

Asymptotic  expression  for  ?zl,  383 

Asymptotic  lines  and  directions,  144 

Asvmptotic  series.  390 

Attraction.  31.  68.  308.  332.  348,  547; 
Law  of  Nature.  31,  307;  motion  tinder, 
190.  264.  .See  Central  Force  and  Po- 
tential 


Average  value,  333 ;  of  functions.  333 ; 

of  a  harmonic  function,  531;  over  a 

surface,  340 
Axes,  right-  or  left-handed,  84,  167 
Axiom  of  contiiuiity,  34 

B.  .See  Bernoulli  numbers,  Beta  fimction 

Bernoulli's  ecjuation,  205,  210 

Bernoulli's  numbers,  448.  456 

Bernoulli's  polynomials,  451 

Bessel's  efjuation,  248 

Bessel's  functions,  248,  393 

Beta  function,  378 

Binomial  theorem,  finite  remainder  in, 
60;  infinite  series,  423,  425 

Binomial,  83 

Boundary  of  a  region,  87.  308,  311 

Boundary  values,  304,  .541 

Brachistochrone,  404 

Branch  of  a  function,  of  one  variable, 
40 ;  of  two  variables,  90 ;  of  a  com- 
plex variable.  492 

Branch  point,  492 

C„.    .See  Cjdinder  functions 

Calculation.  .See  Computation,  Evalua- 
tion, etc. 

Calculus  of  variations,  400-418 

Cartesian  expression  of  vectors.  167 

Catenary,  78,  190;  revolved.  404,  408 

Cauchy's  Formula.  30.  49.  61 

Cauchy's  Integral,  304.  477 

Cauchy's  Integral  test,  421,  427 

Caustic,  142 

Center,  instantaneous,  74,  178;  of  in- 
version, 538 

Center  of  gravity  or  mass,  motion  of  the, 
176  ;  of  areas  or  laminas,  317,  324  ;  of 
points  or  masses,  168  ;  of  volumes,  328 

Central  force,  175,  264 

Centrode,  fixed  or  inovini:.  74 

Chain,  equilibrium  of,  185,  190,  409; 
motion  of,  415 

Chanse  of  variable,  in  derivatives.  12, 
14,  67.  98,  103.  106:  in  differentia! 
equations.  204.  235.  245;  in  inte<rrals. 
16,  21.  54.  65.  328.  330 

Characteristic  curves.  140,  267 

Characteristic  strip,  279 


558 


INDEX 


Charge,  electric,  539 

Charpit's  method,  274 

Circle,  of  curvature,  72  ;  of  convergence, 
433,  437;  of  inversion,  538 

Circuit,  89 ;  equivalent,  irreducible,  re- 
ducible, 91 

Circuit  integrals,  294 

Circulation,  345 

Clairaut's  equation,  230;  extended,  273 

Closed  curve,  308;  area  of,  289,  311; 
integral  about  a,  295,  344,  300,  477, 
536  ;  Stokes's  formula,  345 

Closed  surface,  exterior  normal  is  posi- 
tive, 167,  341;  Gauss's  fornuila,  342; 
Green's  formula,  349, 531 ;  integral  over 
a,  341,  536  ;  vector  area  vanishes,  167 

en,  471,  505,  518 

Commutative  law,  149,  165 

Comparison  test,  for  integrals,  357;  for 
series,  420 

Complanarity,  condition  of,  169 

Complementary  function,  218,  243 

Complete  elliptic  integral,  507,  514,  77 

Complete  equation,  240 

Complete  solution,  270 

Complex  function,  157,  292 

Complex  numbers,  153 

Complex  plane,  157,  302,  360,  433 

Complex  variable.    See  Functions  of  a 

Components,  163,  167,  174,  301,  342,  507 

Computation,  59 ;  of  a  definite  integral, 
77;  of  Bernoulli's  numbers,  447;  of 
elliptic  functions  and  integrals,  475, 
507,  514,  522;  of  logarithms,  59;  of 
the  solution  of  a  differential  equation, 
195.  See  Approximations,  Errors,  etc 

Concave,  up  or  down,  12,  143 

Condensation  point,  38,  40 

Condition,  for  an  exact  differential,  105 ; 
of  complanarity,  169  ;  of  integrability, 
255  ;  of  parallelism,  166  ;  of  perpendic- 
ularity, 81,  165.    See  Initial 

Conformal  representation,  490 

Conformal  transformation,  132,  477,  538 

Congruence  of  carves,  141 

Conjugate  functions,  536 

Conjugate  imaginaries,  156,  543 

Connected,  simply  or  multiply,  89 

Consecutive  points,  72 

Conservation  of  energy,  301 

Conservative  force  or  system,  224,  .307 

Constant,  Euler's,  385 

Constant  function,  482 

Constants,  of  integration,  15,  183;  phys- 
ical, 183  ;  variation  of,  243 

Constrained  maxima  and  minima.  120, 
404 

Contact,  of  curves,  71  ;  order  of,  72  ;  of 
conies  with  cubic,  521  ;  of  plane  and 
curve,  82 

Continuation,  444,  478,  542 


Continuity,  axiom  of,  34 ;  equation  of, 
350  ;  generalized,  44  ;  of  functions,  41, 
88,  476;  of  integrals,  52,  281,  368;  of 
series,  430 ;  uniform,  42,  92,  476 

Contour  line  or  surface,  87 

Convergence,  absolute,  357,  422,  429 ; 
asymptotic,  456 ;  circle  of,  433,  437 ; 
of  infinite  integrals,  352  ;  of  products, 
429;  of  series,  419;  of  suites  of  num- 
bers, 39  ;  of  suites  of  functions,  4.30  ; 
nonuniform,  431 ;  radius  of,  433  ;  uni- 
form, .368,  431 

Coordinates,  curvilinear,  131  ;  cylindri- 
cal, 79;' polar,  14;  spherical,  79 

cos,  cos-i,  155,  161,  393,  456 

cosh,  cosh-i,  .5,  6,  16,  22 

Cosine  amplitude,  507.    See  en 

Cosines,  direction,  81,  169;  series  of,  460 

cot,  coth,  447,  450,  454 

Critical  points,  477,  491  ;  order  of,  491 

CSC,  550,  557 

Cubic  curves,  519 

Curl,  Vx,  345,  349,  418,  -553 

Curvature  of  a  curve,  82  ;  as  a  vector, 
171 ;  circle  and  radius  of,  73,  198 ; 
problems  on,  181 

Curvature  of  a  surface,  144  ;  lines  of,  14'! ; 
mean  and  total,  148;  principal  radii, 
144 

Curve,  308  ;  area  of,  311 ;  intrinsic  equa- 
tion of,  240  ;  of  limited  variation,  309  ; 
quadrature  of,  313  ;  rectitiable,  311. 
See  Curvature,  Length,  Torsion,  etc., 
and  various  special  curves 

Curvilinear  coordinates,  131 

Curvilinear  integral.    See  Line 

Cuspidal  edge,  142 

Cuts,  90,  302,  362,  497 

Cycloid,  76,  404 

Cylinder  functions,  247.    See  Bessel 

Cylindrical  coordinates,  79,  328 

D,  symbolic  use,  152,  214,  279 

I)arboux"s  Theorem,  51 

Definite  integrals,  24,  52 ;  change  of 
variable,  54,  65  ;  computation  of,  77  ; 
Duhamel's  Theorem.  63;  for  a  series, 
451;  infinite,  352;  Osgood's  Theorem, 
54,  65 ;  Tiieorem  of  the  Mean,  25,  29, 
52.  359.  See  Double,  etc..  Functions, 
Infiiute,  Cauchy's,  etc. 

Degree  of  differential  eijuations,  228 

Del,  V,  172,  260,  343,  345,  349 

Delta  amplitude,  507.    See  dn 

De  Moivre"s  Theorem,  155 

Dense  set,  -39.  44,  50 

Density,  linear,  28;  surface,  315;  vol- 
ume, 110,  326 

Dependence,  functional.  129;  linear,  245 

Derivative,  directional.  97.  172:  geo- 
metric properties  of,  7:  infinite,  46; 


I:^^DEX 


559 


logarithmic,  5;  normal,  97,  137,  172; 
of  higher  order,  11,  67,  102,  197;  of 
integrals,  27,  52,  283,  370  ;  of  products, 
11, 14,  48  ;  of  series  term  by  term,  430  ; 
of  vectors,  170;  ordinary,  1,  45,  158; 
partial,  93,  99  ;  right  or  left,  46 ;  The- 
orem of  the  Mean,  8,  10,  46,  94.  See 
Change  of  variable,  Functions,  etc. 

Derived  units,  109 

Determinants,  functional,  129 ;  Wron- 
skian,  241 

Developable  surface,  141,  143,  148,  279 

Differences,  49,  462 

Differentiable  function,  45 

Differential,  17,  64  ;  exact,  106,  254,  300  ; 
of  arc,  70.  80.  131  ;  of  area,  80,  131 ; 
of  heat,  107,  294  ;  of  higher  order,  67, 
104;  of  surface,  340;  of  volume,  81, 
330;  of  work,  107,  292;  partial,  95, 
104 ;  total,  95,  98,  105,  208,  295 ;  vec- 
tor, 171,  293,  342 

Differential  equations,  180,  267;  degree 
of,  228 ;  order  of,  180 ;  .solution  or 
integration  of,  180  ;  complete  solution, 
270^  general  solution,  201,  230,  269; 
infinite  solution,  230;  particular  solu- 
tion, 230;  singular  solution,  231,  271. 
See  Ordinary,  Partial,  etc. 

Differential  equations,  of  electric  cir- 
cuits, 222,  226 ;  of  mechanics,  186,  263  ; 
Hamilton's,  112  ;  Lagrange's,  112,  224, 
413;  of  media,  417;  of  physics,  524; 
of  striniis,  185 

Differential  geometry,  78.  131,  143,  412 

Differentiation,  1;  logarithmic,  5;  of 
implicit  functions,  117;  of  integrals, 
27,283;  partial,  93 ;  total,  95;  under 
the  sign,  281  ;  vector,  170 

Dimensions,  higher,  335 ;  physical,  109 

Direction  cosines,  81,  169  ;  of  a  line,  81 ; 
of  a  normal,  83 ;  of  a  tangent,  81 

Directional  derivative,  97,  172 

Discontinuity,  amount  of,  41,  462  ;  finite 
or  infinite,  479 

Dissipative  function,  225,  307 

Distance,  shortest,  404,  414 

Distributive  law,  151,  165 

Divergence,  formula  of,  342  ;  of  an  inte- 
gral, 352  ;  of  a  series,  419  ;  of  a  vector, 
343,  553 

Double  integrals,  80,  131,  313,  315,  372 

Double  integration,  32,  285,  319 

Double  limits,  89,  430 

Double  points,  119 

Double  sums,  315 

Double  surface  potential,  551 

Doubly  periodic  functions,  417,  486, 
504,  517;  order  of,  487.  See  p,  sn, 
en,   dn 

Dniiamers  Theorem.  28,  63 

I")iuiin"s  indicatrix,  145 


e  =  2.718-.-,  5,  4.37 

E,  complete  elliptic  integral,  77,  514 

A^-function,  62,  353,  479 

E  {<p,  k),  second  elliptic  integral,  514 

c^,e^  4,  160,  447,  484,  497 

Edge,  cuspidal,  142 

Elastic  medium,  418 

Electric  currents,  222,  226,  533 

Electric  images,  539 

Electromagnetic  theory,  350,  417 

Element,  lineal,  191,  231  ;  of  arc,  70, 
80  ;  of  area,  80,  131,  344  ;  of  surface, 
340  ;  of  volume,  80,  330  ;  planar,  254, 
267 

Elementary  functions,  162  ;  character- 
ized, 482,  497  ;  developed,  4.50 

Elimination,  of  constants,  183,  267  ;  of 
functions,  269 

Ellipse,  arc  of,  77,  514 

Elliptic  functions,  471,  504,  507,  511,  517 

Elliptic  integrals,  503,  507,  511,  512,  517 

Energy,  conservation  of,  301  ;  dimen- 
.sioiis  of,  110;  kinetic,  13,  101,  112, 
178,  224,  413  ;  of  a  gas,  106,  294,  392  ; 
of  a  lamiua,  318  ;  potential,  107,  224, 
301,  413,  547  ;  principle  of,  264  ;  work 
and,  293,  301 

Entropy,  106,  294 

Envelopes,  of  curves,  135,  141,  231  ;  of 
lineal  elements,  192 ;  of  planar  ele- 
ments, 254,  267  ;  of  planes,  140,  142; 
of  surfaces,  139,  140,  271 

Equation,  adjoint,  240;  algebraic,  159, 
306,482;  Bernoulli's,  205,  210;  Clair- 
aut's,  230,  273  ;  complete,  240  ;  intrin- 
sic, 240  ;  Laplace's,  524  ;  of  continuity, 
350  ;  Poisson's,  548  ;  reduced,  240 ; 
Piccati's,  250;  wave,  276 

Equations,  Hamilton's,  112  ;  Lagrange's, 
112,  225,  413.  .See  Differential  equa- 
tions. Ordinary,  Partial,  etc. 

E(iuicrescent  variable,  48 

Equilibrium  of  strings,  185,  190,  409 

Equipotential  line  or  surface,  87,  533 

Equivalent  circuits,  91 

Error,  average,  390  ;  functions,  ij/,  388  ; 
mean  square,  390,  465 ;  in  target 
practice,  390 ;  probable,  389 ;  proba- 
bility of  an,  386 

Errors,  of  observation,  386;  small,  101 

Essential  singularity,  479,  481 

p:uler's  Constant,  385,  457 

Euler's  Formula,  108,  159 

Euler's  numbers,  450 

Euler's  transformation,  449 

Evaluation  of  integrals,  284,  286,  360, 
371.    See  Computation,  etc. 

Even  function,  30 

Evolute,  142.  234 

Exact  differential,  106,  254,  300 

Exact  differential  equatiou,  207, 237, 254 


560 


.NDEX 


Expansion,  asymptotic,  390,  397,  456 ; 
by  Taylor's  or  Maclaurin's  Formula, 
57,  305 ;  by  Taylor's  or  Maclaurin's 
Series,  435,  477  ;  in  ascending  powers, 
433,  479  ;  in  descending  powers,  390, 
397,  456,  481  ;  in  exponentials,  4(i-'), 
467  ;  in  Legendre's  polynomials,  400  ; 
in  trigonometric  functions,  458,  405 ; 
of  solutions  of  differential  e<iuations, 
198,  250,  525.  See  special  functions 
and  Series 
Exponential  development,  405,  407 
Exponential  function.    See  a-'-,  e^ 

F,  complete  elliptic  integral.  507,  514 

F(0,  k)  =  sn-i  sin  <p,  507,  514 

Factor,  integrating,  207,  240,  254 

Factorial,  379 

Family,  of  curves,  135,  192,  228  ;  of  sur- 
faces, 139,  140.    See  Envelope 

Faraday's  Law.  350 

Finite  discontinuity,  41,  402.  479 

Flow,  of  electricity,  553  ;  steadv",  553 

Fluid  differentiation,  101 

Fluid  motion,  circulation,  345  :  curl.  340  ; 
divergence,  343  ;  dynamical  equations, 
351;  equation  of  continuitj^  350;  ir- 
rotational,  533;  velocity  iwtential, 
533  ;  waves,  529 

Fluid  pressure,  28 

Flux,  of  force,  308,  348 ;  of  fluid,  343 

Focal  point  and  surface,  141 

Force,  13,  203;  as  a  vector,  173,  301; 
central,  175;  generalized,  224 ;  prob- 
lems on,  180,  204.    See  Attraction 

Form,  indeterminate,  01,  89;  perma- 
nence of.  2,  478;  quadratic,  115, 
145 

Fourier's  Integral.  377,  400.   528 

Fourier's  series.  458.  405,  525 

Fractions,  partial,  20,  00.    See  Kational 

Free  maxima  and  minima,  120 

Frenet's  fornuilas,  84 

Frontier,  34.    See  Bcnindary 

Function,  average  value  of,  333 ;  ana- 
lytic, 304;  complementary,  218,  243; 
complex,  157,  292;  conjugate,  5o(i ; 
dissipative,  225,  307  ;  doubly  periodic, 
486;  F-f unction,  02  ;  even,  30;  Green, 
535;  harmonic,  530;  inteirral.  433; 
odd,  30;  of  a  complex  varialjle,  157; 
periodic,  458,  485  ;  potential,  .301.  See 
also  most  of  these  entries  themselves, 
and  others  under  Functions 

Functional  dependence,  129 

Functional  determinant.  129 

Functional  equation,  45,  247.  252,  387 

Functional  independence,  12'.» 

Functional  relation.  129 

Functions,  series  of,  4.30:  table  of  ele- 
mentary,  102.    For  special  functions 


see  under  their  names  or  symbols ;  for 
special  types  see  below 

Functions  defined  by  functional  equa- 
tions, cylinder  or  Bessel's,  247  ;  ex- 
ponential, 45,  387  ;  Legendre's,  252 

Functions  defined  by  integrals,  contain- 
ing a  parameter,  281,  368,  376;  their 
continuity,  281,  369;  differentiation, 
283,  370  ;  integration,  285,  370,  373  ; 
evaltiation,  284.  286,  371 ;  Cauchy's 
integral,  304  ;  Fourier's  integral,  377, 
460;  Poisson's  integral,  541,  540;  po- 
tential integrals,  546 ;  with  variable 
limit,  27,  53,  209,  255,  295,  298;  by 
inversion.  490.  503,  517;  conjugate 
function,  530,  542;  special  functions, 
Bessel's,  394.  398  ;  Beta  and  Gamma, 
378;  error,  f.  388 ;  E  (0. k).  514 ;  F(,p. k), 
507  ;  logarithm.  302,  300.  497  ;  7)-f  unc- 
tion, 517;  sin-i,  307,  498;  sn-i,  435, 
503;  tan-i.  307,  498 

Functions  defined  by  mapping,  543 

Functions  defined  by  properties,  con- 
.stant,  482  ;  doubly  periodic,  486 ;  ra- 
tional fraction.  483 ;  periodic  or 
exponential.  484 

Functions  defined  by  series,  p-f  unction, 
487  ;  Theta  functions,  467 

Functions  of  a  complex  variable,  158, 
163 ;  analytic,  304,  435 ;  angle  of, 
159;  branch  point,  492;  center  of 
gravity  of  poles  and  roots,  482  ; 
Cauchy's  integral,  304,  477 ;  con- 
formal  representation,  490  ;  continu- 
ation of,  444,  478,  542  ;  continuity, 
158,  470  ;  critical  points.  477,  491 ;  de- 
fines conformal  transformation,  476 ; 
derivative  of.  158,  470;  derivatives  of 
all  orders,  305;  determines  harmonic 
functions.  530  ;  determines  orthogonal 
trajectin-ies.  194  ;  doubly  periodic,  486  ; 
elementary.  1(52  ;  essential  singularity, 
479,  481;  expansible  in  series,  430; 
expansion  at  infinity,  481 ;  finite  dis- 
continuity. 479  ;  integral,  433  ;  integral 
of,  300.  3()0;  if  constant,  482;  if'ra- 
tional,  483  ;  inverse  function,  477  ;  in- 
version of .  543 ;  logarithnuc  derivative, 
482;  nniltiple  valued,  492  ;  number  of 
roots  and  poles.  482;  periodic,  485; 
poles  of,  480  ;  principal  part,  483  ;  resi- 
dues, 480  ;  residues  of  logaritlunic  de- 
rivative, 482;  l^iemann's  surfaces, 
493 ;  roots  of,  158,  482  ;  singularities 
of,  476,  479;  Taylor's  Fornuda,  305; 
uniformly  continuous,  476;  vanishes, 
158,  See  various  special  functions 
and  topics 

Functions  of  one  real  variable,  40; 
average  value  of,  333;  branch  of,  40; 
Cauchy's  theorem,  30,  49  ;  continuous, 


INDEX 


561 


41;  continuous  over  dense  sets,  44; 
Darboux'sTlieorem,  51 ;  derivative  of, 
45  ;  differentiable,  45 ;  differential,  64, 
67;  discontinuity,  41.  462;  expansion 
by  Fourier's  series,  462  ;  expansion  by 
Legendre's  polynomials,  466  ;  expan- 
sion by  Taylors  Formul^i,  49,  55; 
expansion  by  Taylor's  Series,  435 ;  ex- 
pression as  Fourier's  Integral,  377, 
466 ;  increasing,  7,  45,  310,  462 ;  in- 
finite, 41 ;  infinite  derivative,  46  ;  inte- 
grable,  52,  54.  310;  integral  of,  15,  24, 
52  ;  inverse  of,  45 ;  limited,  40 ;  limit 
of,  41,  44  ;  lower  sum,  51 ;  maxima  and 
minima,  7,  9,  10,  12,  40,  43,  46,  75; 
multiple  valued,  40;  not  decreasing, 
54,  310 ;  of  limited  variation,  54,  309, 
462  ;  oscillation,  40,  50  ;  Rolle's  Theo- 
rem, 8,  46 ;  right-hand  or  left-hand 
derivative  or  limit,  41,  46,  49,  462 ; 
single  valued,  40 ;  theorems  of  the 
mean,  8,  25.  29,  46,  51,  52,  359;  uni- 
formly continuous,  42  ;  unlimited,  40  ; 
upper  sum,  51;  variation  of,  309,  401, 
410.  See  various  special  topics  and 
functions 

Functions  of  several  real  variables,  87; 
average  value  of,  334,  340 ;  branch 
of,  90;  continuity,  88;  contour  lines 
and  surfaces,  87;  differentiation,  03, 
117;  directional  derivative,  97;  double 
limits,  89,  430  ;  expansion  by  Taylor's 
Formula,  113;  gradient,  172;  harmonic, 
530;  homogeneous.  107;  implicit.  177  ; 
integral  of^  315,  326.  335.  340;  intt-- 
gralion,319. 327;  inverse.  124;  maxima 
and  minima,  114.  118.  120,  125;  miui- 
max,  115;  multiple-value(1.90  ;  iK)rmal 
derivative.  97  ;  over  various  regions, 
91;  potential.  547;  single-valued,  87; 
solution  of,  117;  space  derivative,  172  ; 
total  differential,  95;  transformation 
by,  131;  Theorem  of  the  Mean,  94; 
uniformly  continuous.  91;  variation 
of.  90 

Fundamental  solution.  534 

Fundamental  theorem  of  aK'ebra.  159. 
306 

Fundamental  units,  109 

Gamma  function,  378;  as  a  product, 
458  ;  asymptotic  expression,  383,  456  ; 
beta  functions.  379 ;  inteirrals  in  terms 
of.  380  ;  logarithm  of,  383  ;  Stiriin-'s 
Fornuila.  386 

Gas,  air,  189 ;  molecules  of  a,  392 

Gauss's  Formula,  342 

Gauss's  Intesral,  348 

gd,  gd-i,  ey,\Q,  450 

General  solution,  201,  230,  269 

Geodesies,  412 


Geometric  addition.  163 

Geometric  language,  33,  335 

Geometric  series,  421 

Geometry.   See  Curve,  Differential,  and 

all  special  topics 
Gradient,  y,  1"^,  301.    See  Del 
Gravitation.    See  Attraction 
Gravity.    See  Center 
Green  Function,  535,  542 
Green's  Formula,  349,  531 
Green's  Lemma,  342,  344 
Gudermannian  function,  6,  16,  450 
Gyration,  radius  of,  334 

Half  periods  of  theta  functions,  468 

Hamilton's  equations,  112 

Hamilton's  principle,  412 

Harmonic  functions,  530;  average  value, 
531;  conjugate  functions,  536  ;  exten- 
sion of,  542 ;  fundamental  solutions, 
534;  Green  Function,  535;  identity 
of,  534;  inversion  of,  539;  maxinmm 
and  mininuun.  531,  554;  Poisson's  In- 
tegral, 541,  546;  potential,  548;  sin- 
gularities, 534 

Helicoid,  418 

Helix,  177,  404 

Helmholtz,  351 

Higher  dimensions,  335 

Higher  order,  differentials,  67,  104;  in- 
finitesimals, 64.  356;  infinites,  (iO 

Ilonidgeneitv,  phvsical,  109;  order  of, 
107" 

Homo<:eneous  differential  equations, 
204,'  210,  230,  23*;,  259,  262.  278 

Homogeneous  functions,  107 ;  Euler's 
Fcinnula,  108,  152 

Hooke's  law,  187 

Hydrodynamics.    See  Fluid 

IIy]:)erbolic  funetiims.  5.  .Sec  cosh,  sinh, 
etc. 

Hypergeometric  series,  398 

Imaginary.  153,  216;  conjugate,  156 

Imaginary  powers.  161 

Implicit  functions.  117-135.  See  .Max- 
ima and  Minima.  Minimax.  etc. 

Indefinite  integral.  15.  53.  See  Functions 

Independence,  functional.  129;  linear, 
245 ;  of  path,  298 

Indeterminate  forms,  61;  L'llospital's 
Rule,  61  ;  in  two  variables,  298 

Indicatrix.  Dupin's,  145 

Indices,  law  of.  150 

Induction,  308.  348 

Inequalities,  36 

Inertia.   .See  Moment 

Infinite.  66  ;  become.  35 

Infinite  derivative.  46 

Infinite  integral.  352.    .See  Functions 

Infinite  product,  429 


562 


INDEX 


Infinite  series,  30,  419 

Infinite  solution,  230 

Infinitesimal,  03  ;  order  of,  03 ;  higher 
order,  04  ;  order  higher,  350 

Infinitesimal  analysis,  08 

Infinity,  point  at,  481 

Inflection  point,  12,  75;  of  cubic,  521 

Instantaneous  center,  74,  178 

Integrability,  condition  of,  255 ;  of  func- 
tions, 52,  308 

Integral,  Cauchy's,  304;  containing  a 
parameter,  281,  305;  definite,  24,  51 ; 
double,  315  ;  elliptic,  503  ;  Fourier's, 
377;  Gauss's,  348;  higher,  335;  in- 
definite, 15,  53  ;  infinite,  352  ;  inver- 
sion of,  490;  line,  288,  311,  400; 
Poisson's,  541;  potential,  540;  sur- 
face, 340  ;  triple,  320.  See  Definite, 
Functions,  etc. 

Integral   functions,  433 

Integral  test,  421 

Integrating  factor,  207,  240,  254 

Integration,  15  ;  along  a  curve,  291.  400  ; 
by  parts,  19,  307  ;  iiy  substitution,  21 ; 
constants  of,  15,  183  ;  double,  32,  320  ; 
of  functions  of  a  complex  variable, 
307  ;  of  radicals  of  a  biquadratic,  513  ; 
of  radicals  of  a  quadratic,  22 ;  of  ra- 
tional fractions,  20 ;  over  a  surface, 
340  ;  term  by  term,  430  ;  under  tlie 
sign,  285,  370.  See  Differential  eijua- 
tions.  Ordinary,  Partial,  etc. 

Intrinsic  equation,  240 

Inverse  function,  45,  477  ;  derivative  of, 
2,  14 

Inverse  operator,  150,  214 

Inversion,  537;  of  integrals,  490 

Involute,  234 

Irrational  numbers,  2,  30 

Irreducible  circuits,  91,  302,  500 

Isoperimetric  problem,  400 

Iterated  integration,  327 

Jacobian,  129,  330.  330,  470 
Jumping  rope,  511 
Junction  line,  492 

Kelvin,  351 

Kinematics,  73,  178 

Kinetic  energy,  of  a  chain,  415;  of  a 
lamina,  318;  of  a  medium,  410  ;  of  a 
particle,  13,  101  ;  of  a  rigid  body,  293  ; 
of  systems,  112,  225,  413 

Lagrange's  ecjuations,  112,  225,  413 
Lagrange's  variation  of  constants,  243 
Lamina,  center  of  gravity  of.  317; 
density  of,  315  ;  energy  of,  318  ;  kine- 
matics of,  78,  178;  mass  of,  32,  310; 
moment  of  inertia  of,  32,  315,  321; 
motion  of,  414 


Laplace's  equation,  104,  110,  52(),  530, 
533,  548 

Law,  Ampere's,  350;  associative,  150, 
105;  commutative,  149,  105;  distrib- 
utive, 150,  105  ;  Faraday's,  350 ; 
Hooke's,  187;  of  indices,  150;  of 
Nature,  307  ;  parallelogram,  154,  103, 
307  ;  of  the  Mean,  see  Theorem 

Laws,  of  algebra,  153;  of  motion,  13, 
173,  204 

Left-hand  derivative,  40 

Left-handed  axes,  84,  107 

Legendre's  elliptic  integrals,  503,  511 

Legendre's  ecjuation,  252  (Ex.  13  5) ;  gen- 
eralized, 520 

Legendre's  functions,  252 

Legendre's  polynonnals,  252,  440,  400  ; 
generalized,  527 

Leibniz's  Kule,  284 

Leibniz's  Theorem,  11,  14,  48 

Length  of  arc,  09,  78,  131,  310 

Limit,  35 ;  double.  89 ;  of  a  (juotient, 
1,  45;  of  a  rational  fraction,  37;  of  a 
sum,  10,  50,  291 

Limitt'd  set  or  suite,  38 

Linuted  variation,  54,  309.  402 

Line,  direction  of,  81,  1()9 ;  tangent, 
81 ;  normal,  90 ;  perpendicular,  81, 
105 

Line  integral,  288, 298, 311.  400  ;  about  a 
closed  circuit,  295, 344  ;  Cauchy's,  304  ; 
differential  of,  2!)1  ;  for  angle,  297 ; 
for  area,  289;  f(n-  work,  293;  in  the 
complex  plane,  300,  497  ;  independent 
of  jiath,  298  ;  on  a  Riemaim's  surface, 
499,  503 

Lineal  element,  191,  228,  231,  201 

Linear  dependence  or  independence, 
245 

Linear  differential  etpiations,  240  ; 
Bessel's,  248;  first  order,  205,  207; 
Legendre's,  252  ;  of  physics.  524  ;  par- 
tial, 207,  275,  524;  second  order,  244; 
simultaneous,  223 ;  variation  of  con- 
stants, 243  ;  with  constant  coefficients, 
214,  223,  275 

Linear  operators,  151 

Lines  of  curvature,  140 

log,  4,  11,  101,  302,  449,  497 ;  log  cos,  log 
sin,  log  tan,  450  ;   —  logr,  535 

Logaritlnnic  differentiation  and  deriv- 
ative, 5 ;  of  functions  of  a  complex 
variable,  482  ;  of  gamma  function, 
382  ;  of  theta  functions,  474,  512 

Logarithms,  computation  of,  59 

Attest,  432 

Maclaurin's  Formula,  57.    Sec  Taylor's 

^laclaurin's  Series,  435 

Mii^iiitude  of  complex  numbers,  154 

Mapping  regions,  543 


IIs^DEX 


563 


Mass,  110;  of  lamina,  316,  32;  of  rod, 
28;  of  solid,  32(5;  potential  of  a, 
308,  348,  527.    See  Center  of  gravity 

Maxima  and  minima,  constrained,  120, 
404 ;  free,  120 ;  of  functions  of  one  vari- 
able, 7,  9, 10,  12,  40,  43,  46,  75 ;  of  func- 
tions of  several  variables,  114,  118, 120, 
125 ;  of  harmonic  functions,  531  ;  of 
implicit  functions,  118,  120,  125;  of 
integrals,  400,  404,  409  ;  of  sets  of  num- 
bers', 38  ;  relative,  120 

Maxwell's  assumption  for  gases,  390 

]Mayei''s  method,  258 

Mean.    See  Theorem  of  the  Mean 

Mean  curvature,  148 

Alean  error,  390 

Mean  square  error,  390 

Mean  value,  333,  340 

Mean  velocity,  392 

Mechanics.  See  E(iuilibrium,  Motion, 
etc. 

Medium,  elastic,  418;  ether,  417.  See 
Fluid 

Meusnier's  Theorem,  145 

IMinima.    See  Maxima  and  minima 

Minimax,  115,  119 

Minimum  surface,  415,  418 

Modulus,  of  complex  number,  154;  of 
e*lliptic  functions,  k,  k',  505 

Molecular  velocities,  392 

Moment,  176;  of  momentum,  176,  264, 
325 

Moment  of  inertia,  curve  of  mininuim, 
404;  of  a  lamina,  32,  315,  324;  of  a 
particle,  31 ;  of  a  solid,  328,  381 

Momentum,  13,  173;  moment  of,  176, 
264,  325;  principle  of,  264 

Mongers  method,  276 

Motion,  central,  175,  264;  Hamilton's 
equations,  112;  Hamilton's  I'rinciple, 
412;  inaplane,  2r)4  ;  Lagrange's  equa- 
tions, 112,  225,  413;  of  a  chain,  415; 
of  a  drumhead,  52(i ;  of  a  dynamical 
system.  413  ;  of  a  lamina,  78, 178.  414  ; 
of  a  medium,  41(5 ;  of  the  simple  pen- 
dulum, 509 ;  of  systems  of  particles, 
175 ;  rectilinear,  186 ;  simple  harmonic, 
188.   -See  Fluid,  Small  vibrations,  etc. 

INIultiple-valued  functions,  40,  90,  492 

Multiplication,  liy  complex  numbers, 
155;  of  series.  442  ;  of  vectors,  164 

Multiplier,  474;  undetermined.  411 

Multipliers,  method  of.  120,  126,  406, 
411 

^lultiply  connected  regions,  89 

Newton's  Seconil  Law  of  Motion,  13, 173, 
186 

Normal,  principal,  83;  to  a  closed  sur- 
face, 167.  341 

Nori  lal  derivative,  97,  137,  172 


Normal  line,  8,  96 

Normal  plane,  181 

Numbers,  Bernoulli's,  448 ;  complex, 
153  ;  Euler's,  450  ;  frontier,  34  ;  inter- 
val of,  34  ;  irrational,  2,  36 ;  real,  33  ; 
sets  or  suites  of,  38 

Observation,  errors  of,  386 ;  small  er- 
rors, 101 

Odd  function,  30 

Operation,  149 

Operational  methods,  214,  223,  275,  447 

Operator,  149,  155,  172  ;  distributive  or 
linear,  151;  inverse,  150,  214;  invol- 
utorv,  152  ;  vector-differentiating,  172, 
260, '343,  345,  349 

Order,  of  critical  point,  491 ;  of  deriv- 
atives, 11 ;  of  differentials,  67  ;  of 
differential  equations,  180;  of  doubly- 
periodic  function,  487  ;  of  homogene- 
ity, 107 ;  of  inhnitesimals,  63 ;  of 
intinites,  66;  of  ix)le,  480 

Ordinary  differential  equations,  203 ; 
approximate  solutions,  195,  197;  aris- 
ing from  partial,  534  ;  Bernoulli's,  205, 
210;  Clairaut's,  230;  exact,  207,  237; 
homogeneous,  204,  210,  230,  236  ;  inte- 
grating factor  for,  207;  lineal  element 
of,  191;  linear,  see  Linear;  of  higher 
degree,  228;  of  higher  order,  234;  prob- 
lems involving,  179;  Kiccati's,  250; 
systems  of,  223,  260 ;  variables  sepa- 
rable, 203.    See  Solution 

Orthogonal  trajectories,  plane,  194,  234, 
266';  space,  260 

Orthogonal  transformation,  100 

Osculating  circle,  73 

( )sculating  plane.  82, 140.  145,  171,  412 

Osgood's  Theoi'em,  54,  65,  325 

p-f unction.  487,  517 

Pappus's  Theorem,  332,  346 

Parallelepiped,  volume  of,  169 

Parallelism,  condition  of,  166 

Parallelogram,  law  of  addition,  154,  163, 
307;  of  periods,  486;  vector  area  of, 
l(i5 

I'arameter,  135  ;  integrals  Avith  a,  281 

Partial  derivatives,  93 ;  higher  order, 
102 

Partial  differentials,  95,  104 

Partial  differential  equations,  267;  char- 
acteristics of,  267,  279;  Charpifs 
method,  274 ;  for  types  of  surfaces. 
269;  Laplace's,  526;  linear,  267,  275, 
524  ;  ]\Ionge's  method,  276  ;  of  physics, 
524 ;  Poisson's,  548 

Partial  differentiation,  93.  102  ;  change 
of  variable,  98,  103 

Partial  fractions,  20,  06 

Particular  solutions,  230,  524 


664 


INDEX 


Path,  independency  of,  298 

Pedal  curve,  9 

Period,  half,  468 ;  of  elliptic  functions, 

471,  480;  of  exponential  function,  161; 

of  theta  functions,  468 
Periodic  functions,  161,  458,  484 
Pennanence  of  form,  2,  478 
Physics,  differential  equations  of,  524 
Planar  element,  254,  267 
Plane,  normal,  81 ;  tanfrent.  96 ;   oscu- 
lating, 82,  140,  145,  171,  412 
Points,  at  infinity,  481;  consecutive,  72  ; 

inJiection,  12,  75, 521 ;  of  condensation, 

38,  40  ;  sets  or  suites  of,  380  ;  singular, 

119,  476 
Poisson's  equation,  548 
Poisson's  Integral,  541 
Polar  cooi'dinates.  14,  79 
Pole,  479;  order  of,  480  ;  residue  of,  480  ; 

principal  part  of,  483 
Polynomials,   Pernoulli's,  451  ;    Leuen- 

dre's,  252,  440,  466,  527  ;  root  of,  159, 

482 
Putential,  308,  332,  348.  527,  530.  539, 

547  ;  double  surface,  551 
Potential  energy,  107,  224,  301,  413 
Potential  function,  301,  547 
Potential  integrals,  546;  retarded,  512; 

surface,  551 
Power  series,  428,  433,  477  ;  descending, 

389,  397,  481 
Powers  of  complex  numbers,  161 
Pressure,  28 
Principal  normal,  83 
Principal  part.  483 
Principal  radii  and  sections,  144 
Principle,  Hamilton's,   412 ;  of  energy. 

264  ;  of  momentum,  264  ;   of  moment 

of   momentum,  264 ;  of   permanence 

of  form,   2,  478  ;  of  work  and  enerirv, 

293 
Probability,  387 
Probable  error,  389 
Product,    scalar,    164;   vector,    165;    of 

complex  numbers,  155;  of  operators, 

149;  of  series,  442 
Products,  derivative  of,  11,  14,  48;  in- 

tinite,  429 
Projection,  164,  167 

Qaadratic  form,  115,  145 

Quadrature,  313.    See  Integration 

Quadruple  integrals,  335 

Quotient,  limit  of,  145;  of  differences. 
30,61;  of  differentials.  64,  67;  of  power 
series,  446;  of  theta  functions,  471 

Paabe"s  test.  424 

Radius,  of  convergence,  433,  437;  of  cur- 
vature. 72,  82,  181;  of  gyration,  -V-'A  ; 
uf  torsion,  83 


Hates,  184 

Katio  test,  422 

Rational  fractions,  characterization  of, 
483 ;  decomposition  of,  20,  66  ;  inte- 
gration of,  20  ;  limit  of,  37 

Keal  variable,  35.    See  Functions 

Rearrangement  of  series,  441 

Rectifiable  curves,  311 

Reduced  equation,  240 

Reducibility  of  circuits,  91 

Regions,  varieties  of,  89 

Relation,  functional,  129 

Relative  maxima  and  mimima,  120 

liemainder,  in  asymptotic  expansions, 
390,  398,  4.56;  in  Taylor's  or  Mac- 
laurin's  Formula,  55,  306,  398 

Residues,  480,  487  ;  of  logarithmic  de- 
rivatives, 482 

Resultant,  154,  178;  moment,  178 

lietarded  potential,  552 

Reversion  of  series,  446 

Revolution,  of  areas,  346 ;  of  curves, 
332;  volume  of,  10 

Rhumb  line,  84 

Riccati's  equation,  250 

Niemann's  surfaces,  493 

Right-hand  derivative,  46 

Right-handed  axes,  84,  167 

Rigid  body,  energy  of  a,  293;  with  a 
fixed  point,  76 

Holle's  Theorem,  8,  46 

Roots,  of  complex  numbers,  155 ;  of 
polynomials,  156,  159,  306,  412;  of 
unity,  156 

Railed  surface,  140 

Saddle-shaped  surface,  143 

Scalar  product,  164.  168,  343 

Scale  of  numbers.  33 

Series,  as  an  inteirral.  451  ;  asvmptotic, 
390,  397,  456;  binomial,  423.  425; 
Fourier's,  415;  infinite,  39.  419;  ma- 
nipulation of,  440 ;  of  complex  terms, 
423;  of  functions,  430;  Taylor's  and 
Maclaurin's,  197,  435,  477;  theta, 
467.   See  various  special  functions 

Set  or  suite,  38,  478  ;   dense,  39,  44,  50 

Shortest  distance,  404,  412 

Sigma  functions,  a,  ca,  523 

Simple  harmonic  motion,  188 

Simple  pendulum,  509 

Simply  connected  region,  89.  294 

Simpson's  Rule.  77 

Simultaneous  diiferential  equations.  223. 
260 

sin.  sin-i.  3.  11.  21,  155,  161.  307,  43(  . 
453,  499 

Sine  amplitude.  507.    Sec  sn 

Single-valued  function.  4().  !-i7,  2'.t5 

Singular  jioints.  119.  476 

Singular  solutions,  230,  271 


INJ)EX 


5G5 


Singularities,  of  functions  of  a  complex 
variable,  476,  479  ;  of  harmonic  func- 
tions, 534 

sinh,  sinh-i,  5,  453 

Slope,  of  a  curve,  1  ;  of  a  function,  301 

Small  errors,  101 

Small  vibrations,  224,  415 

sn,  sn-i,  471,  475,  503,  507,  511,  517 

Solid  angle,  347 

Solution  of  differential  equations,  com- 
plete, 270  ;  general,  260 ;  infinite,  230  ; 
particular,  230,',524  ;  singular,  230,  271 

Solution  of  implicit  functions,  117,  133 

Speed,  178 

Spherical  coordinates,  79 

Sterling's  approximation,  386,  458 

Stokes's  Formula,  345,  418 

Strings,  equilibrium  of,  185 

Subnormal  and  subtangent,  8 

Substitution.    See  Change  of  variable 

Successive  approximations,  198 

Successive  differences,  49 

Suite,  of  numbers  or  points,  38  ;  of  func- 
tions, 430 ;  uniform  convergence,  431 

Sum,  limit  of  a,  36,  24,  51,  419;  of  a 
series,  419.  See  Addition,  Definite  in- 
tegral. Series,  etc. 

Superposition  of  small  vibrations,  226, 
525 

Surface,  area  of,  67,  339 ;  closed,  167, 
341;  curvature  of,  144;  developable, 
141,  143,  148,  279;  element  of,  340; 
geodesies  on,  412;  minimum,  404,  415  ; 
normal  to,  96,  341;  Riemann's,  403; 
ruled,  140 ;  tangent  plane,  96 ;  types 
of,  269;  vector,l67;  w-,  492 

Surface  inteitral,  340,  347 

Symbolic  methods,  172,  214,  223,  2()0, 
275,  447 

Systems,  conservative,  301;  dvnamical, 
413 

Systems  of  differential  e(iuations,  223, 
260 

tan,  tan-i,  3,  21,  307,  450,  457,  498 

Tangent  line,  8,  81,  84 

Tangent  plane,  96,  170 

tanh,  tanh-i,  5,  0,  450,  501 

Taylor's  Formula,  55,  112,  152,  305,  477 

Taylor's  Series,  197,  435,  477 

Taylor's  Theorem,  49 

Test,  Cauchy's,  421;   comparison,  420; 

Raabe's,  424;  ratio,  422;  Weierstrass's 

M-,  432,  455 
Test  function,  355 
Theorem  of  the  Mean,  for  derivatives. 

8,  10.  46,  94 ;  for  integrals,  25,  29,  52 

359 
Thermodynamics,  106,  294 
Theta  functions,  //,  H^.Q.  6j,  as  Fourier's 

series,  467;  as  products,  471  ;  define 


elliptic  functions,  471,  504;  logarith- 
mic derivative,  474,  512  ;  periods  and 
half  periods,  468  ;  relations  between 
S(]uares,  472  ;  small  thetas,  d,  da,  523  ; 
zeros,  469 

Torsion,  83;  radius  of,  83,  175 

Total  curvature,  148 

Total  differential,  95,  98,  105,  209, 
295 

Total  differential  equation,  254 

Total  differentiation,  99 

Trajectory,  196;  orthogonal,  194,  234, 
260 

Transformation,  conformal,  132,  476; 
Euler's,  449 ;  of  inversion,  537 ;  orthog- 
onal, 100;  of  a  plane,  131;  to  polars, 
14,  79 

Trigonometric  functions,  3,  161,  453 

Trigonometric  series,  458,  465,  525 

Triple  integrals,  326  ;  element  of,  80 

Umbilic,  148 

Undetermined  coefficients,  199 

Undetermined  nmltiplier,  120,  126,  406, 

411 
Uniform  contiiuiity,  42.  92,.  476 
Uniform  convergence,  369,  431 
Units,  fundamental  and  derived,   109; 

dimensions  of,  109 
Unity,  roots  of,  156 
Unlimited  set  or  suite,  38 

Vall^e-Poussin,  de  la,  373,  555 
Value.    See  Alisolute,  Average,  Mean 
Variable,  complex,   157;    eciuicrescent, 

48 ;  real,  35.  See  Change  of.  Functions 
Variable  limits  for  integrals,  27,  404 
A'ariables,     separable,    179,    203.     See 

Functions 
Variation,  179;  of  a  function,  3,  10.  54; 

limited,  54.  309;  of  constants,  243 
Variations,  calculus  of,  401  ;  of  integrals, 

401,  410 
Vector,  154, 163;  acceleration,  174;  area, 

1()7,  290 ;  components  of  a,  163,  167, 

174,   342;    curvature,    171;    moment, 

176;    moment    of    momentum,    ]7(); 

momentum,    173  ;    torsion,    83,    171  ; 

velocity,  173 
Vector   addition,    154.    163 
Vector  di fferential ion,  1 70, 260, 342, 345  ; 

f  (n-ce,  1 73 
Vector  functions,  260,  293,  300,  342,  345. 

551 
A'ector  operator  y.  see  Del 
A'ector  product,  l(i5,  168,  345 
Vectors,    addition    of,    154,    163 ;    com- 

planar,  169;    nudtiplication    of,    155. 

163  ;  parallel,  l(i6  ;  perpendicular,  165  ; 

products  of,  164,  165,  168,  345;  pro- 
jections of,  164,  167,  342 


566 


IXDEX 


Velocity,  13,  173 ;  anjjular,  346 ;  areal, 
175  ;  of  molecules,  392 

Vibrations,  small,  224,  526 ;  supeiposi- 
tion  of,  226,  524 

Volume,  center  of  gravity  of,  328 ;  ele- 
ment of,  80  ;  of  parallelepiped,  169  ; 
of  revolution,  10 ;  under  surfaces,  32, 
317,  381 ;  with  parallel  bases,  10 

Volume  integral,  341 

Wave  equation,  276 
Waves  on  water,  529 


Weierstrass's  integral,  517 

Weierstrass's  J/-test,  432 

Weights,  333 

Work,  107,  224,  202.  301 ;  and  eneriry, 

293,  412 
Wronskian  determinant,  241 

z-plane,  157.  302,  360,  433;  mapping 
the,  490.  497,  503,  517,  543 

Zeta  functions,  Z,  512  ;  f,  522 

Zonal  harmonies.  See  Legendre's  poly- 
nomials 


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